The Volga, yes. Stalingrad and its particular portion of it, not necessarily. That is a rather small region to cram in the entire striking force of the drive to the south, and if the Germans just go there and stop, the Soviets will begin a series of counterattacks above and more slowly below Stalingrad. This is the 1942 version of the drive only to Moscow. The claim that the Soviets need the Volga region to produce tanks is somewhat belied by Stalingrad's gutting in the OTL battles and this not imairing the USSR, as well as this view relying on a rather certain neglect of a difference between Russian geography and industrial power and Soviet.
The overwhelming majority of Lend-Lease came in through the Pacific, not Persia. Nor is it exactly clear how a narrow, hammering attack focused purely on the Stalingrad region cuts off Soviet access to the south, as per the requirements of the OP. The Germans had troops as far south as Ordzikhondize IOTL, and were able to cut their way straight through the Volga twice in the course of the Stalingrad battles of OTL to no effect. All this also ignores that even focusing purely on Stalingrad Germany's means were far too underwhelming for the scale of the task set to them in the usual pattern. Of course I get that the idea that the Nazis can somehow win the War in the East has no regard for logistical or strategic realities of this sort, relying instead on vague statements involving butterflies and total handwaving of certain all too vital realities of this particular war.