It's not just about running game logic. The CPU has a lot to do with getting the game to run at 2x the framerate. The only reason CPUs havent been the bottleneck in the PC space is because games were designed with shitty jaguar CPUs in mind and then ported to PC where getting the game to run at higher framerates became a GPU limitation.
Yes, the CPU has a lot to do with getting the game to run at 2X
the frame rate, but nothing to do at all with getting the game to run at 2X
the resolution (if the rest of the setting are the same, of course, since some settings can indeed have some impact on the CPU). If the CPU is able to run a game at just 30 fps dropping the resolution won't help, it'll still be a 30 fps game.
If you want to play a 30 fps game at 60 fps you need 2X the CPU power and 2X the GPU power. Buy if you just want to play the game at 2X the resolution you just need a beefier GPU, a stronger CPU with the same GPU won't give you a higher rez. That's why many PS4 Pro games run at 2X the resolution while its CPU it's only slightly faster than that of base PS4, because its GPU is 2X faster (a little more in fact). And that's also why while Pro can run games at 2X the resolution it can't run the games at 2X the frame rate unless the game was heavily bottlenecked by the GPU.
On the other hand Series S runs circles around One X CPU and games are usually higher rez on One X. It just has a stronger GPU and much higher bandwidth, which is what matters for higher resolutions. As you can see having 4 times the CPU power doesn't help Series S.
This is easily benchmarkeable in the PC world.