There is correct information in this video, except there is one important factor this guy overlooked when talking about being limited to a 60hz display, you still don't get the visual fluidity of 200fps if your monitor can only refresh 60 times per second.
And yes it does affect the responsveness of your game, because it means you have a tougher time tracking movements of your opponents. That's why competitive gaming/esports monitors generally start at 120hz, not 60hz. So in the end, it's still a waste of time pursuing beyond 60fps if your monitor is incapable of refreshing more than 60 times per second, it is not up for debate. If you want to take full advantage of 120+fps then it makes sense to invest in a proper gaming monitor, if you buy garbage, you get garbage, period.