The only way (or so it seems to me) that the Russians could have maintained a fighting force in Stalingrad was by resupplying them across the Volga.  If the far side of the river was captured territory, Stalingrad would be a trapped pocket.  The fighting forces inside would not have been able to reform a cohesive fighting force under constant artillery bombardment and the greater part of the German army could have been free to deal with actual threats to itself.  I think if the 6th Army had crossed the river and fought the Russians on the other side, away from urban areas,  they could have maintained momentum and kept the Russians from regrouping. 
Any way you look at it, military action in urban (or any other civilian populated areas) ends badly for everyone involved (especially the civilians whose homes are destroyed, families and friends killed, etc...) and should probably be avoided if at all possible.