Your average AAA power fantasy FPS isn't going to function well with a disadvantaged PC, but how about horror games? They are not quite up my alley, but I would assume that they have weaker PCs to amplify the horror aspects. How do games like Silent Hill and the like function in this regard?
Generally I feel like horror games avoid progression because it's gamey and that detracts from the horror factor. (The more abstract things that are distracting you from how scary the game is, the less scary it is. Mechanics in a horror game are in a way an expression of power that is in direct contrast to how dangerous the world is. You instantly feel safer in a horror game when you get a gun, for example, or when you get abilities that let you circumvent threats in the horror narrative.)
So yeah. Silent Hill, you don't get more powerful. Amnesia, you don't get more powerful.
My thinking is that convincingly giving the player someone weak, who seems human and fallible, right down to the animations and sound effects, makes the journey of becoming powerful that much more meaningful. I think it'd work in a FPS, given the right approach to balance.
I mean imagine you're basically playing a fat nerd, who sprints and you can hear him wheezing, saying stuff like "God I'm so out of shape." Fast forward an hour into play, he's not wheezing anymore. He's lost some weight. He runs faster, for longer. His dialog is filled with a rising confidence in himself. By the end of the game you can stand and look back at where you started and actually feel the progression on more levels than just the intellectual/mathematical.
That'd be my hope anyways. It's ultimately more of an immersion thing than anything else, something grounded in a story that mimics what's going on mechanically.
I'd point to GTA:San Andreas as something that played with this concept. While it wasn't balanced for long-term play, I did enjoy watching my character change and improve.