It absolutely is an inherent problem of anti-cheat software. By definition, that software exists to deny you the freedom to cheat or otherwise modify the game. The only way they can possibly do that is to pwn your machine.

Don't generalize that to all software. Free software doesn't do that. GNU was literally founded on the principle of not doing that.

That’s not a “problem” with anti-cheat, it’s the entire point. It is a group of people who have chosen to give up their freedom to modify their games in exchange for the ability to play with other people who agree to the same.

It is okay for private clubs to have rules for the conduct of their consenting members.

FOSS is great for allowing individual freedom of software developers. It doesn’t provide functional solutions for all of the worlds problems, though.

They don’t prevent you to cheat, they monitor to detect cheat. Big difference.

Plenty of software will kick you from servers or even kill the game process if they detect anything, including false positives like some suspicious string in a completely unrelated chat application.

Who cares though? Fine, you can say they monitor you. The point is: to monitor you, they gotta pwn your machine. If they don't you can just defeat their silly "monitoring" software. They own your machine by installing kernel mode malware into your OS. They literally rootkit your computer.