There is still theoretically an edge if you just show the same frames statistically earlier to one player.

You can present the game state statistically earlier to the player with the higher refresh rate display.

My theory about why it helps gamers to have a higher frame rate is that for something like a whip turn, with a low frame rate, your brain has to take a brief moment to work out where it ended up looking after the pan. But if your frame rate is high enough, you brain can keep updating its state during the pan because the updates are continuous enough not to lose “state” during it. This means when you finish the fast move, there is no delay while you reorient yourself for a few milliseconds.

Current esports titles don't use tick rates.

https://youtu.be/GqhhFl5zgA0

You can film the screen in slow motion and visually see more fluid motion (and see how it reacts to player input).

Games also use predictive methods and client side hit detection to mitigate most of the effects of network latency in the common cases.

Yes, because the visual feedback on your client matters as well (faster reaction times).

Also, it means each frame is more recent. Also, higher refesh rates reduce the mean variance between the timestamps of rendered frames offered by the GPU that are drawn on the monitor, meaning the visuals are smoother. The smoothness is pleasing to the eye and makes it easier to focus.