You have some misconceptions.
Fps is limited by the cpu, not the gpu. Cpu can only pre-render so many frames per second, depending on the game code itself. In CSGO on an i7-3770K I get 300fps average. The gpu has to live upto that limit or not, depending on the resolution and detail settings. I run a gtx970 and CSGO is ultra max everything at 1080p. Still get 300fps. If I drop to low settings, I get 300fps, because that's the limit of frames per second the cpu can pre-render. Gtx970 is capable of higher, but is limited by the cpu, so changing detail settings does nothing.
That would change if I enabled 4k DSR, and rendered at 4k resolution in the gpu, max fps would drop, cpu still capable of 300fps, gpu not, so lowering settings would get more fps.
But that has nothing to do with the monitor. The monitor can only show you as many frames per second as it has Hz refresh, depending on what the gpu gives it. So if the gpu gives it 50fps, that's what you see, no matter if it's 60Hz, 144Hz or 240Hz.
With CSGO at 300fps, it's beyond the refresh of any monitor, so all I'd get is Hz, just no dips below Hz. With Skyrim, all I get is 60fps, cpu limited by mods, so I'd get 60fps on 60Hz, 144Hz or 240Hz.
Having fps higher than Hz means no reliance on vsync or its alternatives to balance frame dips.