so several things wrong again.

If your frames are set to max even without a 240hz monitor the input lag will be unnoticeable

just false. its easily noticeable on slower refresh monitors. and i do mean easily.

If someone with a 240hz monitor and someone with a 144hz monitor look in each others direction at the exact same time, their reaction speeds aren't fast enough to capitalize on something that takes place in about 2 server ticks.

in competitive shooters specifically, thats not how things tend to work. person A is holding an angle, person B is going to peek that angle. having a 240hz monitor will allow you to see the person several ms faster than having a 144hz monitor. this is the definition of a competitive advantage. if the players have similar reaction times, then every little bit makes a difference. if person A sees person B several ms faster, he will also be able to shoot several ms faster. if you dont consider this a competitive advantage, then i suggest looking into what a competitive advantage actually is.