Because of flanks. That's why the Germans didn't do it. Their flanks were already vulnerable enough. If they tried to surround Stalingrad, then not only will their northern and southern flanks will be horribly exposed, but the Germans who would have held the eastern banks of the Volga would be fighting the Soviets forces still in Stalingrad, AND the Soviet forces in the eastern side of the Volga.

And the Germans didn't even have enough troops to fully capture Stalingrad, that's why they had to rely mainly on their weaker Axis allies to guard their flanks. Laying a siege generally means that you have an overwhelmingly numerical superiority over your enemy. The Germans had no such thing.