The city is located on both sides of the river, although the bigger part is on the West bank. The city is a port, an important port like Danzig, only much less capable for actually shipping large volumes. But it of course controlled shipping up the Oder. If following the river strictly, that would mean the city would have to be divided. Not ideal, but then not unheard of either as Berlin, Frankfurt/Oder and Görlitz demonstrate. Only that the existing port and really everything important is located entirely in the West of the city.
We can be sure that in these discussions some crazy voices would come up with history: Despite being predominantly settled by Germans for centuries and being administratively part of German states since the 13th century, it might be a bit stretchy argued that it was during the 10th and 11th century somehow Polish. Somehow as this is a nationalistic concept and quite anachronistic to describe the situation on the ground in these terms. But the then established border looks interesting:
It might be a bit unfair to only ask for "why did Poland actually want". It should be quite clear that there was no official Poland when that decision was advanced for the first time. Nothing would move on that front without the Soviet Union. The Soviets themselves reinstated a provisional German administration in the Western parts of the city at first. Seeing that some Polish chauvinists demanded an ever far reaching expansion to the West since the 19th century does not mean that this was a position of any majority or considerable influence when Stalin had the last word on everything. Some demands published around the Potsdam conference even included a cordon sanitaire of at least 10km West of the left riverbank of the Odra, bringing cities like Forst, Görlitz, Frankfurt in their entirety under Polish administration.
The timeline to observe here is that in 1941 Stalin advanced the idea of annexing parts of Poland East of the Curzon-line and proposed a territorial compensation for Poland carved from German territory. During the Potsdam conference the Western Polish border was then defined with words that clearly left Stettin under Soviet occupation, not Polish. But Stalin insisted repeatedly that Stettin would be Polish.
For the above analysis refer to Clemens Heitmann: "Die Stettin-Frage
Die KPD, die Sowjetunion und die deutsch-polnische Grenze 1945", Zeitschrift für Ostmitteleuropa-Forschung 51 (2002) H. 1.
After the end of the war, the exact course of the demarcation line between the Soviet occupation zone of Germany and the German territories in the Szczecin region under Polish administration was still unclear, so that the Red Army initially refrained from handing over Szczecin, located west of the Oder, to the Polish authorities. It established a newly formed German administration in the city, initially from 2 May 1945 under Erich Spiegel as mayor, who was replaced on 26 May 1945 by the KPD politician Erich Wiesner. On July 5, 1945, however, Szczecin was handed over to Poland by the Soviet commandant (see also the Schwerin Border Treaty). (Allied agreements provided for a border "directly west of Swinemünde and from there along the Oder to the confluence of the western Neisse", the Oder-Neisse line. At the same time, the German city administration was replaced with the deposition of Mayor Wiesner, and the settlement of Poland began, which accompanied the expulsion of the German civilian population. The newly settled inhabitants had belonged in part to the Polish ethnic minority in areas east of the Curzon Line that had fallen into the Soviet Union. Szczecin was rebuilt as the capital of the voivodship of the same name and with the reactivation of industry, educational institutions, etc. The port was only handed over to Poland by the Soviet Union in 1955.
— WP: Stettin
But there was indeed a Polish voice advancing the annexation of Stettin with hinterland as a whole –– and as compensation. Compensation for losses to the Soviets in the East and of course for the effects of the German invasion, occupation and destruction (of lives and material).
Who advanced this idea? The provisional Polish government under Borkowicz appointed the city-planner Piotr Zaremba to go there and seize an opportunity that was indicative for a slowly growing agreement between Polish and Soviet side.
While not playing a role in Yalta, the reinforced Polish position did play a role during the Potsdam conference:
The position of the People's Poland authorities at the Potsdam conference
In July 1945, the Provisional Government of National Unity issued a memorandum on the issue of the western border of Poland, which is a reference point for the Polish position during the Potsdam conference. It lists the following arguments for territorial changes:
Compensation - moral compensation for the treatment of the entire nation as a society of farmhand and material compensation for plundering and destruction of national wealth. Poland, being the ally of the victorious states, has the right to demand in the name of all-human justice for its attitude, losses and sacrifices, to obtain compensation in kind, in labor workshops, i.e. in the land, factories, mines, cultural property, and above all in territory corresponding to the needs of the nation.
Demographic needs of Poland. The Polish nation can not develop normally if it is forced to emigrate at the same time, therefore taking into account: 1) high birth rate, 2) demand for emigration, and 3) need of re-emigrants, Poland must have an area that meets the needs of its population […] Only the shift of the border to the Odra and Nysa rivers provides opportunities for satisfying the most important demographic needs of the nation. In agricultural terms, Poland was the most overpopulated country in Europe. As a result of territorial changes, this overcrowding may be mitigated in the case of the inclusion of the territory after the Odra and Nysa.
Polishness of the western territories: Apart from Polish borders from 1918 to 1939, significant areas inhabited by the majority of the Polish population remained.
Historical Polish Rights: History from the 10th century shows that Poland stood firmly on the Oder […], and in certain periods it significantly exceeded it. It was only from the thirteenth century that it began to lose its western frontiers under German pressure.
Geographical location and geopolitical relations of the western territories with Poland. Geographical location is the basis of the policy of a given country. Poland also has to adapt to this law. Poland lies on the eastern reaches of Western Europe. It lies on the interstate piers: Adriatic-Baltic and Baltic-Black Sea, through which for a long time have been running trade routes that have maintained their economic and political importance to this day. There are also two transcontinental routes running through Poland: lowland and submontane, they connect with each other capitals and industrial centers of a number of European countries. Geographical location predisposes all Poland to take the role of an intermediary in Europe between West and East and South and North […]
Poland strives for its natural boundaries, which are: in the south Sudety and Karpaty, in the north coast of the Baltic Sea, and in the west Odra and Nysa. The area thus designated creates a compact geographical unit, connected with the Vistula and Odra basins, directed by natural roads towards the Baltic Sea.
Economic relations of the western territories with Poland. The area of eastern Germany, inhabited mostly by Poles, from the second half of The nineteenth century could not feed and keep its population in place. Almost all of the natural growth was forced to emigrate to Western Germany or abroad. The economic potential of the Polish population was constantly decreasing under the influence of the tendentious Prussian policy. East Prussia and Silesia artificially detached from Poland, from their natural base, did not show normal development. Silesia, Oder, Szczecin, cut off from Poznań and Pomerania since 1918, could not develop either. It was particularly severe for Silesia, deprived of the eastern Polish markets. As a result, almost the whole area of eastern Germany was an émigré area […].
The connection of these lands with Poland will allow, on the one hand, employment of surplus rural population in industry, on the other, it will provide the industry with a convenient and extensive market. Within the Reich, these lands were a deficient borderland and did not have the possibility of proper development, while industrial production was focused mainly on exports (see Appendix No. 4). On the other hand, within the Polish state, the industry will find very favorable conditions, working mainly for the internal market.
At the Potsdam conference itself, the Polish side was represented by Bolesław Bierut , Edward Osóbka-Morawski , Stanisław Grabski , Stanisław Mikołajczyk , Wincenty Rzymowski and Michał Rola-Żymierski .
In the speech of Deputy Prime Minister Mikolajczyk, who supported the proposed territorial changes, there was also the argument that they are a guarantee of preventing the basic causes of German imperialism by taking away the industrial base that Germany used to develop its military potential. Thus, as Prime Minister Mikolajczyk said, these changes will not only help to rebuild the destruction of Poland, but will contribute permanently to maintaining world peace and will contribute to the security of all nations.
— WP: Konferencja poczdamska (machine translated)
A governmental study for the US concluded:
[…] “we should resist the exaggerated claims now being advanced by the Provisional Government in Lublin for ‘compensation’ from Germany which would include the cities of Stettin and Breslau in Poland and make necessary the transfer of from eight to ten million Germans.
This action would have made any new Polish state dependent on the Soviet Union for protection of its borders against a discontented German population unless the West specifically granted such a guarantee, an unlikely event in the face of Roosevelt’s concern about maintaining Allied unity. In any case, this was not a totally new claim from the Lublin Poles. Ambassador Harriman had notified the secretary of state in a telegram from Moscow on 19 December 1944 that a very real possibility existed that the Soviet-backed Poles would demand this territory. His warning showed great insight; he urged the secretary to register some type of protest before the United States was confronted with a fait accompli, the exact situation that developed a few months later.
Third, the Yalta briefing book went on to analyze possible reasons for the Lublin government’s insistence on these “exaggerated claims.” The authors speculated the following:
Poland, which would depend completely on Moscow for protection from German irredentism after acquiring a large part of German territory and transferring 8 to 10 million Germans, might become a “full-fledged” Soviet satellite;
If the proposed world security organization proved impossible, it would be to the Soviet advantage to have a Polish border as far west as possible;
By giving as much land as possible to the Poles in the west, the loss of 43 percent of its land in the east might not be felt quite as greatly.
The matter was of course far from "settled" then:
Tension was high after Stalin objected to the proposal that he claimed had been made by the president and Churchill at Yalta; namely, that Poland’s western border should be along the Oder and eastern Neisse Rivers since this border would leave the towns of Stettin and Breslau in German hands. Stalin then pointed to the line that he favored on a map and told the Westerners that the issue involved was one of frontiers and could not be ignored. Finally, Churchill withdrew his objection to having the Polish government send representatives to be heard by the foreign ministers at Potsdam.[…]
Invitations were issued to two or three representatives of the Polish Provisional Government; but at least eight came, including Bolesław Bierut, president; Edward Osóbka-Morawski, premier; Właydsław Gomułka, deputy prime minister and minister of western territories; and Stanisław Mikołajczyk, deputy prime minister and minister of agriculture. They met with the foreign ministers on 24 July 1945 to present their views. Bierut postulated that the new border in the east would require the surrender of 18,000 square kilometers of land but would make Poland more ethnically homogeneous and was thus justified. The proposed border in the west along the Oder and western Neisse (with Stettin included) was important for security and economic reasons. Bierut here contradicted Stalin’s earlier claim that all the Germans had fled the area when he contended that the 1.5 million remaining Germans would probably be willing to leave if the area were administered by Poland. The further advantage for Poland would be that Poles would no longer have to emigrate for jobs or sell their labor in this area while others made profits.
The next month Lane requested permission and funds to open a consulate in Stettin (Szczecin), since it would become an important port serving the Oder River and its tributaries. In his report to the department he noted that the Soviet army occupied a large part of the area and that the Polish authorities had begun to provide essential services to the city. Lane viewed the city as an important post for reporting political activity that could not be covered by the officers in Warsaw and emphasized again the “good effect” that U.S. consulates in Poland could have on U.S.-Polish relations. Acting Secretary Dean Acheson denied Lane’s request, not on diplomatic grounds concerning the ambiguity of Stettin’s future but because of funding difficulties. He appointed two more officers to the Warsaw office who could be temporarily detailed to Stettin and notified Lane that “consideration will be given [to the] possible establishment [of a] regular office [there].
Lemmer responded to this by pointing out that the lands gained from Germany in the west were much more fertile than those lost to the Soviets in the east, implying that the Poles needed less land. The Poles then presented their suggested border, after conceding that the existing line resulted in an underpopulated Poland and an overpopulated Germany. The proposed compromises, if they were meant in earnest, might have surprised the Polish officials’ superiors in Warsaw. They demanded retention of the estuary of the Oder River, including Stettin; the Oder would still serve as the border, but the regions of Lower and Central Silesia where they lie west of the Oder would be negotiable. In response to Lemmer’s question, the Poles indicated that the industrial region of Waldenberg (in Lower Silesia) was more important to the Germans than to the Poles and that this region—even up to the Neisse River—was open to discussion. Murphy’s speculation that the Poles had arranged this meeting, probably with Soviet backing to address rumors circulating about a potential border adjustment, reveals that he did not know the degree to which the Soviets were in control of the situation.
In March 1953 N. Spencer Barnes, chief of the Eastern Affairs Division at HICOG Berlin, analyzed press reports from Stockholm claiming that the administration of Stettin (Szczecin) would be given to the East Germans. Barnes wrote that although HICOG had no proof of such a change, it “would not be illogical” since the city was an important commercial port for East German barges, and he continually heard reports about Polish resistance against the Soviets in the area. He added that such a rumor did not necessarily mean that supervision of Stettin would be given to the East Germans but suggested that the Soviets might make some move to raise the status of the “‘more reliable’” East Germans. “In view of the tentative arrangements on the OderNeisse boundary in the Potsdam Agreement,” Barnes wrote, “such developments may warrant careful observation.”
— Debra J. Allen: "The Oder-Neisse Line : The United States, Poland, and Germany in the Cold War Contributions to the Study of World History", Greenwood: Westport: London, 2003. (Instructive review)
Some of the above had as a consequence to the effect that in terms of investment and development, the Western parts of Poland didn't receive much love at first. It might look like Poland actually didn't want Stettin and hinterland.
Much of what might be seen as hanging in the balance after 1949 has to be seen as rumours and tactic for some negotiations. But after the Stalin-Note and the solidification of Germanies not into a single state but two in opposing military blocks after 1955 the matter really started to remain as stable as it is now. Since 1990 no German state claims the city of Stettin for itself.