Question about FPS

If I do not use V-Sync, my game reaches to 200 or sometimes 300 FPS, I have heard some pro players like to keep V-Sync off in order to make the game reacts faster to their input.

So for a 60Hz monitor, if we run the game at 200FPS how does it make you feel smoother?

I understand people are saying the game is processed 144 times per second, so if we move the mouse, the game should react within 1/144 seconds. but if we use V-Sync, in order to eliminates screen tearing the game would be locked at 60 FPS same as monitors refresh rate. Then the game will react within 1/60 seconds, that sounds a lot of slower, I get that....

But for a online game like lol, there is a always a latency, I bet everyone here have experience that if the latency goes up to 500ms, you can clearly feel that your game is reacting to your control half second late... So this shows that latency is clearly effecting the user input react time for a online game...(I don't know how the game handles it maybe send the movement to the server then make it move until receive server's response back?). Or some other games with different handling method you would experience you just did the move, but next second it comes back to where you were, because of the latency is too high.

So here is what I do not understand: if the game could have the latency as low as 20ms(not sure how you guys are thinking but I think this is pretty good for me). Then due to the latency, the game cant react faster then 1/(1000ms/20ms) second, this is only about 1/50 second, so how could 144 FPS helps here, it cant even reach to 1/60.... Can anyone please explain this to me, or I am totally thinking at the wrong direction?... Thank you.