It'll still be better than 99.99% of human beings at clicking on heads, and will still win you 99% of fights against non-cheaters.
Now what do you do, in your correctly coded game?
Keep in mind, as a cheat developer, I can keep moving these goal posts.
(As an aside, we already have games using built in assistance/cheats as an accessibility feature to level the playing field for people using certain hardware or controllers e.g. games with PC and console crossplay, so perhaps there's some approaches and lessons from that which we can apply here as well.)
The only thing at stake then is the integrity of leaderboards or tournament brackets, which is a problem that already had some different, separate approaches to solve. But IMO that a problem is absolutely subservient to the integrity and playability of the base game.
"... that a problem is absolutely subservient to the integrity and playability of the base game."
If you use hacks/cheats to help you play higher than your natural skill/ability, doesn't that ruin the integrity of the game? The mission is not accomplished in this case. Instead it's just tricky to determine who's hacking.
Headshotting everyone on-sight also sounds like the old days of hacks in games. In my experience, there is more nuance to the way these cheats are used these days. Most people aren't trying to be obvious about it.
Do you have any experience developing networked multiplayer games? I do it for a living, and it's not that easy, at all. In a situation where I shoot at your head on my screen, but you were already around the corner on your screen, what happens? This happens _all_ the time due to latency, and has nothing to do with development costs or infra. There's no way to avoid the problem when you have people with varying levels of connection quality, and different distances to the game server.