High-privilege anti-cheat software will always require high privileges to install (though it does not necessarily need any special privileges to run the game after installation). Linux is no more protected than Windows here. If a game installer installs a kernel module on Linux, that has just as many permissions and can do just as much damage if untrustworthy as a similar kernel module on Windows. What's more, if Linux ever takes off for gaming, you'll see such anti-cheat kernel modules appearing (for e-sports games, at least) for the same reason that they exist on Windows: if you don't have them, the cheat code will go into the kernel where user-mode anti-cheat software can't find it.
One possible difference is that, on Linux, distributing software that links directly into the kernel but is not compatible with the GPLv2 is frowned upon, and the kernel will complain about being tainted with proprietary software. Of course, if you're playing games on Linux at all, there's a decent chance your kernel is already tainted by the proprietary NVidia graphics driver. Also, it's not like the kernel has some magic way to tell whether any given module is open source or not; modules can lie to the kernel about that, and/or after being loaded, they can modify the "am I tainted?" check and flag within the kernel to report whatever they want. Whether such lies or modification would violate the GPL is a question for the courts to decide, but it wouldn't be the first time that proprietary software has pulled such a stunt.
(Or you could make the anti-cheat software open source, but nobody is likely to do that; if the cheat makers can see exactly what the anti-cheat is doing and when, it's way easier to evade or spoof it, and thus immediately win the current round of the cheat-vs-anticheat battle.)
Obviously, cheat-vs-anticheat is a constant game of cat-and-mouse. Anti-cheat software can only look for the kinds of cheats it knows about. Cheat software can - especially if installed before the anti-cheat - modify the anti-cheat software or the kernel itself such that the anti-cheat thinks it's running but can't see the cheat software. The anti-cheat software could potentially try to go even deeper - to the hypervisor, to the firmware, to the hardware (or rather to firmware running on peripheral hardware such as the GPU or network card) - but it can't ever go deeper than the cheat software can in theory go, so there's no way to reliably "win" this contest.
Whether or not cheat vs. anti-cheat is winning in the moment, all of us who don't cheat lose. Installers for games (that we probably don't even play competitively) worm their way deeper and deeper into the system, changing (and potentially breaking or spying on or opening backdoors in) stuff that the user doesn't even know about. Game publishers spend resources on anti-cheat systems instead of gameplay improvements, and OS developers have to accommodate third-party code in unexpected places doing unexpected things, or else people complain that their games stopped working after the last OS update. It's deeply unfortunate.