That leaves the Soviets in control of the major industrial center and regional rail and river transport hub, anchoring their own defense and providing them a bridgehead for their own offensive operations in the future. So long as the Soviets hold Stalingrad they can continually threaten to strike southwest from it down the major rail lines that lead from Stalingrad to Rostov at the mouth of the Don river, through which the main German logistical route passed. If they do so, the Germans have to withdraw from the Caucasus or face the prospect of having all the troops there cut off. Basically, taking Stalingrad and neutralizing the threat it poses, is a necessary prerequisite for continued German operations south and east of the Don.
In fact, in late August of 1942 when the Axis were still securing their line at the Don, the Soviet Stalingrad Front launched several attacks down those routes, punching across the Don and securing several bridgheads that the Germans were unable to eliminate. They would have continued such attacks had not the Germans attacked the city themselves, forcing the Front onto the defensive. Basically, Stalingrad is a continual dagger aimed at the logistical neck of Army Group A, and the Germans do indeed need to take it.
Stalingrad was never going to surrender to a bombardment, and it couldn't be properly seiged since the Soviets controlled the other side of the river. And the longer you leave it in Soviet hands, the more powerful forces the Soviets can build behind it and then cross through it and their other bridgeheads. Failing to take it means that the Axis have a continual existential threat on the left flank of their own push for the Caucasus. Not having Sixth Army balls deep in the city when the Soviet counter offensive launches may allow it to retreat intact, but in the face of the Soviet winter offensive the Germans are likely still going to have to fall back to a more defensible line at the Don river or in Ukraine.
If you admit you don't have the forces to take Stalingrad and neutralize the threat it poses, then you're tacitly admitting that the whole of Fall Blau is being set up for eventual failure, and then why are you continuing to waste resources on ANY operations , defensive or offensive, east and south of the Don?