Frame challenge. The German positions, even at deepest penetration, were undefendable.
...the Germans held portions of the west bank both north and south of the city on which they could place artillery and air units to disrupt Volga River traffic.
Was there any time prior to the encirclement that the Germans' military objectives would have been substantially fulfilled by consolidating their holdings and "ring fencing" the remaining portion of the city while they wen about their business (like setting up a defense)?
The easiest part first, there were no airfields that far forward that were useful (or, indeed, useable) to affect Volga traffic. The airfields that could be used were the same that were later used to (attmpt to) support the Stalingrad pocket, 50 or 100 kilometers back. You don't want to rearm bombers on the runway while in range of the enemy's artillery. So airfields were a non-issue either way. What about ground troups positioning?
North of the city, a small corridor of about 10 kilometers width was held since late August 1942 by German XIV Panzerkorps. This did interrupt the railway lines from Stalingrad to Moscow; that much was achieved. However, this corridor was through open steppes. The Don (which further west provided a defensible landmark for the Romanian 3rd Army curves to the south of Stalingrad.
This northern corridor was under continuous attacks from the north by Soviet 66th Army, 4th Tank Army, 1st Guards Army, and 24th Army. These attacks were of a magnitude that XIV Panzerkorps was unable to assist with attacks on Stalingrad. Actually, LI Armeekorps' advance on Stalingrad from the West was temporarily halted, its support redirected north, to reinforce XIV Panzer Korps.
Only two to three weeks later (around September 11th) German forces were able to close the pocket around Stalingrad in the south. However, this corridor (held by XXXXVIII Panzerkorps) was also under significant attacks by Soviet 64th Army from the south.
All the while, the units of Soviet 62nd Army fighting in Stalingrad proper were getting reinforcements from the eastern bank of the Volga.
Any German units on the west bank of the Volga -- be that in the northern corridor (XIV Panzerkorps), the southern corridor (XXXXVIII Panzerkorps), or in the city proper (LI Armeekorps) -- had Soviet forces on two sides (not counting artillery fire across the river). Quite close as well, no more than 5 kilometers at the most. Not the position you would want to dig into for defense, let alone bring your artillery into. Especially since the terrain outside of the city is nothing but flat steppes
Note that 62nd army was continuously reinforced across the river, which should give an indication that river traffic was not really "controlled" at that point.
German troops were being attacked on all sides. There was no position that could be made into a viable defensive line that could be held through the winter. As Operation Uranus proved in November, which did not target the German spearhead at all, but its exposed flanks hundred miles and more to the west and south.
The advance on Stalingrad was a spearhead, its grip on the west bank of the Volga tenuous, its flanks under massive pressure. Winter was fast approaching, which the Germans had learned meant the upper hand for the Russians for a couple of months. Those encircling corridors did little toward the military goals (other than interrupting the railway to Moscow), could not have been extended to do more (that is what the Germans were trying), and could not have been held as-is during the winter.
A great week by week visualization of the Battle of Stalingrad is this video by Indy Nidell. Watching it should make it clear how tenuous and short-lived the hold to the Volga was.