It really depends on the game.
I believe the most proeminet case and what popularized the "FPS thing" was Counter Strike.
Those studies that say eye can't more than 60Hz may be right in theyr context but for gaming this isn't the whole truth. Using CS GO as an exemple:
Game capped at 60FPS and monitor refresh rate set at 60Hz will get you no where in todays style of competitive play. You can have some fun but unless you are totally gifted, forget high tier competitive play.
The game engine is quite good and the more FPS the less input lag and smoother game, even with the monitor at 60hz.
With higher refresh rates, like 144 or 240Hz, we can see the animations smoother and hit much better, I've test this in the past myself.

I'll say this: who ever said more than 60fps or more than 60Hz doesn't matter never played CS-GO competitevelly. Today where most people have pc capable of doing 300fps and many people use 144hz monitors, if using low end stuff simply cant compete with that.

The big problem comparing the "Hz thing" is that monitors are quite intricate pieces of technology and is very difficult to compare directly 60 vs 144Hz, since other stuff is really important as well, like who is testing, response time varies with monitor (even varies with the same monitor at different settings), input lag (the same) etc.... And until today I´ve never seen a good test comparing this, either the tester don't understand games or don't understand technology.

Some people are not as sensible to Fps as others. The most important is the fps doesn't varies during gameplay, other wise it create jerkiness, input lag and small freezes. Playing a game at 30Fps locked is much enjoyable than playing a game that floats around 40fps, going up to 50/60 and drop to 30 sometimes. Playing at locked 30 usually gives more input lag but you kinda get used to it after some time.

Of course, all this depends on the game and I'm pretty much talking about SINGLE player games here (even CSGO I've tested it on single player) Multiplayer it really depends on the server and every one's connection to it.

On "into the breach" I don't even know what fps it runs and I don't care, mouse seem smooth enough.
On "ori and the blind forest", "Guacamelee" and driving games, like "Dirt rally", if it drops constantly from 60fps I prefer to play 30fps locked.
"Skullgirls", "brawlhalla" and other fighting games, anything below 60fps is a no go.
"CS-GO" minimum ~200fps and 144Hz monitor to play competivelly

On a final note, V-Sync creates a HUGE input lag.