At the end, the stronger player will be the one with biggest computing power.
Advantages you cannot influence (or worse: only influence by investing a lot of money) make games become not fun very quickly.
Currently "proof of work" in shooters is how fast you can click on a moving pixel on the screen. Imagine someone with a faster computer running faster in these games.
You would need to "outsource" proof of work to the player - and input devices are also not that optimal, for example if you require to have something typed in, someone who is a fast typer has a real advantage and others might find this game rally hard and annoying.
Yes, you could implement various attacks/defenses etc. but then it gets complicated again.
Also you would kill a lot of computational power just to be able to not trust the persons whose game you bought and are playing...