Worth noting that even in the FPS genre, that sort of PvE gameplay is actually the original gameplay style, that got subsumed by the massive popularity of PvP shooters later on.
Early FPSes like Wolfenstein, Doom, and the Marathon trilogy were primarily about killing asymmetric AI enemies—and even sometimes (moreso in the Marathon games) progressing through a story.
Doom had deathmatches, where you fought each other instead of the enemies. PvP in general is as old as FPS multiplayer (talking about PvE in single-player is redundant).
I guess I see that more as single-player vs multiplayer rather than PvE vs PvP. But I think your point is that the emphasis shifted from an entirely developer-created experience to one where other players are a major/dominant part of the experience. This shift didn't take that long, though, with multiplayer-only games becoming prevalent (Tribes, Unreal Tournament, Quake III Arena, etc.) even before the 1990s were out, and single-player mode becoming a vestigial element even in those games that still included it before the end of the next decade.
Oh, absolutely. I wasn't trying to claim that the shift in FPSes to primarily-PvP didn't happen until the ubiquity of always-on broadband made internet matchmaking a fully viable thing, just that even in FPSes the roots of the genre were single-player, and even included some fairly prominent story-heavy titles.
Early FPSes like Wolfenstein, Doom, and the Marathon trilogy were primarily about killing asymmetric AI enemies—and even sometimes (moreso in the Marathon games) progressing through a story.