Did it matter that the Red Army clung on to Stalingrad's West bank in 1942?

Was it necessary? No. Many Russians viewed it as symbolically important, since the Volga is considered the boundary of Europe and Asia.

I keep hearing people say that if the Germans had not made certain mistakes they would have 'won' WW II. However, the mistake was starting war in the first place. Within Europe, they were already suffering resistance from Britain, Poland, the Balkans, some parts of Greece, and the communist underground in France. In short, they were creating committed resistance movements as fast as they 'conquered', therefore a limited number of Germans were trying to run vast swaths of territory inhabited by hostile locals. The involvement of the US finished them off quickly, but it would have happened soon enough in any case.

The western bank of the Volga in Stalingrad distracted attention and dispersed the strength of the Wehrmacht. In the same way, strikes from the north of Stalingrad distracted the attention of the divisions. Wehrmacht was forced to destroy the bridgehead and defend Stalingrad from the north and south. As a result, all the shock divisions concentrated in one place. Checkmate.

p.s. "shock divisions" - 4TA.

Yes. Even though I agree with both @Tom and @Kunikov's answers, I feel that there's also another element.

Let's pretend Uranus, the encirclement phase, never happened at all. Even then the sacrifice of the 62nd Army is worthwhile.

The Germans have so far achieved a lot by using Blitzkrieg and encirclement tactics. They've used their armor and superior mobility to flow around Russian armies tasked by Stalin with point defence, rather than meeting them head-on where the Russians are strong. Disproportionate Russian losses aren't so much the result of direct battle casualties as what happens when they get cut off: Viazma means 500K prisoners, for example.

Stalingrad, up to November, is not that kind of battle. It's a vicious urban battle, where the Panzers' mobility is wasted and the infantry is put through a meatgrinder. The Volga, which is nearly 1km wide at this point, keeps the Germans from encircling the city and allows the Russians to drip-feed troops into it at their initiative. Russian courage and unwillingness to retreat, which have so far been a liability in terms of avoiding annihilation by being surrounded, become a strength, precisely due to the geography.

Even now, modern armor-heavy armies tend to shun urban combat whenever possible. The Russians are trading casualties at a lot less disfavorable rate than have been anywhere so far. They have the men to lose, Germany doesn't, so you can think of it as a proto-Kursk in a way: anything not obviously a German tactical victory is in a fact a German strategic defeat.

Second, I am not sure what the rest of Russian front is up during that period. I know the Germans mess around in the Caucasus and they are still besieging Leningrad. But, from August to October, much of the allocatable German effort is at Stalingrad. Certainly, that's where the Luftwaffe is concentrated. So that means that 40% of the 5 month fighting season, May to October, is spent by the Germans at Stalingrad rather than doing much else.

Had the 62nd retreated to the East bank (and it does seem less built-up, btw*) then the German army could have disengaged, avoided their massive losses through November and been used elsewhere. Even without Uranus, Stalingrad up to November is still not a very clever move by the Germans and a very worthwhile effort by the Russians.

* Google Maps

The first stage of the Stalingrad campaign (pre-encirclement), lasted from mid-September to late middle November, or just over three months. The implications would have been very different if the Russians had lost the west bank on say, September 15th, October 15th, or November 15th.

If the Germans had captured the west bank of the city by about November 15th, the result would hardly have been different. The encirclement attack would have proceeded, and captured the German Sixth Army.

If the Germans had captured the west bank by October 15th, the encirclement attack might not have succeeded. That is, the Germans would have had a chance to rest, fortify Stalingrad in safety, and reorganize their troops, particularly their mobile divisions, to face the west and guard their flanks with the Volga at their "backs." In the actual battle, some of the heaviest, and probably the most critical casualties, took place just after the "final offensive" beginning October 14th.

If the Germans had captured Stalingrad by September 15th (which they would have if the Soviets hadn't rushed reinforcements into the city), this victory might by itself forestalled the encirclement campaign. In any event, the Germans might have even enjoyed follow up successes, such as moving down the Volga, capturing Astrakhan, and cutting off oil supplies from the Caucasus.

My best source on the course of the battle is Chuikov's "The Battle For Stalingrad." The interpretations are mine, not his.