Yeah, but the PPSh-41 (as well as its predecessor, the PPD-40 and its successor, the PPS-43) was built in a time before the assault rifle existed in the first place. Back then you either had a submachine gun or you had a gigantic bolt-action rifle with a low capacity and way more power and range than any soldier would actually need. The assault rifle was meant to bridge the gap by totally replacing both weapons in Soviet service with non-specialist soldiers, as it had relatively low recoil and just as much volume of automatic fire as an SMG while still hitting harder than a pistol cartridge.
Most other nations actually kept building SMGs (this is when the "second generation" of submachine gun design came into vogue everywhere, with ultra-cheap guns made of stamped and spot-welded steel with cheap plastic grip panels; the "first generation" was the relatively expensive, heavy stuff made with fancy wooden furniture derived from rifles like the MP-18 and the "third generation" is where we are now, with PDWs that can punch through armor and Picatinny rails and strong plastic receivers), but the Soviet Union tried to use the AK to replace them almost entirely.
The Skorpion, for instance, was almost totally restricted to vehicle crews and got most of its pop culture reputation from use by criminals and guerrillas; they even built the AKS-74U exclusively as a crew weapon before the special forces realized how useful such a compact carbine would be. The PM-63 I believe was exclusively restricted to Poland, and most communist nations that built SMGs were generally Warsaw Pact countries that built them independently and kept them to themselves. The USSR did build a new design, the OTs-02 Kiparis, which was designed in the 1970s but never actually entered service until the 90s after the Soviet Union collapsed. Only in more recent years has Russia started to embrace the submachine gun as a general service weapon in the age of assault rifles.
Even then, they want to overdo it: they made their own custom loads for the 9x19mm round that increase its power to try and make it punch through body armor better, and most of their current 9mm weapons are built stronger than normal so they can easily fire such a powerful cartridge.