How were German soldiers chosen to be flown out of Stalingrad?

The German high command (i.e. Hitler) first refused to believe that Stalingrad would fall and then grasped at straws that this would be a favorable loss ratio and hence worthwhile. The official radio broadcast was "the sacrifice of this army was not futile."

But German propaganda did not admit that many troops surrendered. The official line was that the army "died fighting to the last breath." The promotion of General Paulus to Generaldfeldmarschall was understood as a sign that he should commit suicide since a German Generalfeldmarschall being captured was considered impossible. He didn't take the hint.

A few casualties (genuine and fake) got out. For a general, it would have been difficult to get reassigned in an administrative sleight of hand. They were too visible.

For the morale purposes

Big part of Third Reich ethos and National-Socialist ideology was sense of duty. To put it simply, Volk und Vaterland were above everything. It was expected from everybody to do its utmost to defend them, even sacrificing its own life.

Third Reich has been proud to be based on Führerprinzip, i.e. on the idea that certain gifted people are born to lead. This "born leaders" theoretically should be best of the best, true meritocracy and aristocracy. It was expected from this elite to be able to inspire men under their command, in this case troops in the field. And since commanders demanded unquestionable loyalty and obedience, they too had to share fate of their troops. In the end, this even applied to the Hitler himself, as he also chose to remain in Berlin till the end and commit suicide, rather than let's say attempt to flee to South America.

Germans indeed often ridiculed generals of enemy armies that abandoned their troops. One such example was MacArthur's escape from Philippines . Goebbels mocked this by calling him a "fleeing general", for Mussolini he was a coward. From German generals it was expected something entirely different - even surrender was somewhat shameful. Paulus was promoted to Generalfeldmarschall with implicit expectation he would commit suicide, since no German field marshal was ever captured alive before that.

In this particular case (Stalingrad), a lot has been written about what Germans could have done. Was breakout possible, how much would it cost, were they sacrificed for vain prestige of holding Stalingrad, or by tying up Soviet troops they saved retreating Army Group A etc ... Without going into debate over this, it could be said that German aerial effort had both morale and practical purpose. Germans were striving to evacuate wounded, those who already did their duty to fatherland but were practically unable to fight anymore. Specialists who were no longer unable to do their duty (for example tank crews without tanks), at least not to the fullest, were also sometimes flown out. Generals who still had troops to command were not flown out, for already mentioned morale reasons, and because they still could function inside the pocket. Of course, as always, there were exception to these rules, with chaos on the ground and people simply abandoning high morale principles to save their own skin.

As for Rommel, he was actually relived of command on March 9th 1943, before complete Axis collapse in North Africa. His successor von Arnim did remain with the troops and surrendered to the British on May 12th 1943. It was speculated that Rommel was purposefully removed from command as situation was getting dire to avoid another incident like with Paulus, which happened only few months ago. Rommel was much better known and liked in Germany than Paulus, so his capture would be even heavier blow for the morale.