A 144 Hz monitor would have a refresh period of 7 ms, and a response time of...whatever, doesn't matter. Typical human reaction time is in the range of a few hundred milliseconds, depending on the task, so reducing frame updates from 16 ms (60 Hz monitor) to 7 ms is not that great, a 5% savings assuming best-case.
Here's a colorful plot from the paper Increasing Speed of Processing with Action Video Games by Dye. One thing to note is that when people say "reaction time", it's really this vague, catch-all bin of things. It can include spatial cueing, the most basic see-and-respond test (maybe if you were staring at a corner waiting for a bad guy to come around) which takes 200-300 ms, or harder things like visual searching which involve scanning a scene to detect some feature (maybe a bad guy hiding in a corner), which takes in excess of a second.
Anyways, the graph shows different types of tests (colors/legend) under different conditions (each individual point), given to two groups (the two axes). Video game players (VGPs) perform about 10% faster than novice video game players (NVGPs) with no loss of accuracy.
Ultimately, being better at any video game comes down to training through repetition, where you eliminate mental steps like "grenades...so hit G..." to just thinking and doing, as well gaining a better understanding of the world and its myriad interactions. People can attain "millisecond reaction times", but only through prediction, gained through experience.