I agree that one unified stat seems limiting, but even 1 more would add lots of simple depth. Lots of binaries to choose from (physical/mental, detachment/attachment, instigation/reaction, etc.) that I think could be used to still represent the ways in which people are affected by a sort of crumbling world. For the issue of being good at either one or the other, I see at least one stupid but perhaps worth considering option: roll the stats anyways. In those classic games like Gamma World, it was possible for certain characters to be simply better than others. I think this was built on the understanding that everyone was probably gonna die pretty soon regardless, so you'd get a good chance at rolling a powerful character as well in time. And, it feels good to roll well. But aside from that, it's sort of down to how obstacles are set up, right? Like maybe it's often beneficial to roll middling instead of too high or low, which would benefit the middling statted player. I struggle with the same thing in some of my games, so I don't have any particularly sweet answer.
I think that building it around the narrative tropes of apocalypse movies is a great idea. After all, these are story engines and there's a reason those story beats resonate consistently. And even in their own movies, the tropes are the result of certain unspoken rules in those movies. For example, when you're kicked out of the safe place, it's probably because if had you been the person that did the kicking, you might have chosen the same thing. Every man for themself type deal. So you could recreate those scenarios without writing that it has to happen- only make it mechanically imperative for things to go that way. Want to stay in the nice city? Pay high taxes. If you can't afford them, you're kicked to the scorching desert. Either the players go on an adventure finding work to pay, or they can't and go on an adventure finding a new home. Kind of a win-win for drama (given there's fun rules for both cases).
It looks to me like there's something down them roads.