You seem to overestimate the pure fact of the loss of the Goumrak airfield (on 21st of January), as even after that the 6th army still got the supplies by air (by another small airfield near Goumrak and by using parachutes) until the very end (which is not the end of January 1943, but exactly the 2nd February 1943, when the Strekker's group (ca. 40000 men) had capitulated).
It seems that virtually all sources agree that the Germans in Stalingrad got the following daily air-cargos:
25-29th Nov. - 53,8 tons per day
1-11th Dec. - 97,3 tons per day
13-21st Dec. - 137,7 tons per day
23 Dec. - 11th Jan. - 105,45 tons per day
12-16th Jan. - 60 tons per day
17-21th Jan. - 79 tons per day
22-23th Jan. - 45 tons per day
24 Jan. - 2nd Feb. - 77,9 tons per day
So the main problem was rather the effectiveness of the Soviet air-blockade (i.e. the increasing casualties in the Luftflotte 4), plus the advances and operations against the external air-fields and bases (such as Raid on Tatsinskaya), which eventually made air distances to rise from 200 to 450 kilometers, and finally the advance of 10th-14th January which led to the capture of the main air-field at Pitomnik.
Thus, the Germans in Stalingrad could never "run out of ammo" literally. However, the amount of supplies even in December 1942 was too low compared to the real needs of the 6th army. And an average ratio was probably even close to 1/10th, as per the prognose given by von Weichs in his report to Hitler on 22nd Nov. 1942.
Did it make sense for the Germans to shoot off the last of their ammunition, and then surrender Blücher-style a week before they actually did?
Well, what is sense and how many lives it's equal to? And why it is only "a week before"? In fact, the operation "Ring" costed a lot for the Soviet army, so Paulus' resistance cannot be called insignificant. Nonetheless, many people suggest that Paulus should have accepted the ultimatum on the 8th January. For example, Hans Doerr in his "Der Feldzug nach Stalingrad" says that after the 22nd of December (the fail of the "Winter Gewitter") Paulus had the perfect right to disobey Hitler's orders, and already on the 8th of January the 6th army was literally "in the same situation as Bluecher in Ratekau".