I notice if I am running below 60 - especially in FPS games. Even 59 have the "wrong feel" to it. Getting as low to 40 fps makes me reconsider currently selected graphic options. Especially in multiplayer games. Need. Those. Frames. Seriously.
Same in RTS titles, below 60 fps I can feel already not only the smoothness of things on the screen being wrong but also my control getting worse because of that. As somebody who played StarCraft II for 9 years on "ok-ish" level, that would be absolutely distruptive to any competitive gameplay.
So in short: yeah I do care about my fps and I don't play anything that I can't run at at least stable 50-60 fps. (looking at RDR2 and muttering "your time will come" under his breath)
Yeah, that's why I keep mentioning 'competitive shooters' as an exception, @SIET_Arduous . Fighting games are an even more extreme example. DOA5 actually has a hard limit of 60fps specifically to keep the playing field level.
What gets crazy is when people take that thinking into other games, where it simply doesn't matter that much. The less that precise reaction time matters, the less frame rates matter. When you get all the way down to turn based games like Civilization or XCOM, you just need enough to make the animations and screen scrolling look nice.