It does not matter what games you are running or whether they have anti-cheat. If you are running a program without any sandboxing, they can access everything on your system that your user has access to.
Steam launches games (doesn't matter if you use Proton or they are running natively) as a subprocess and if Steam itself is not sandboxed, those games will also not be. While the symlinks may be abused for this, Wine also gives access to running apps by mounting your Linux root folder as the Z drive (similar to C or D drives you may have already seen).
Running any untrusted software on your system (particularly proprietary ones like most games), you should be mindful of associated privacy risks.
I also wanna address one thing about the particular games you mentioned: EAC and BattlEye run with elevated privileges on Windows. This means it can access and modify things on your system that your normal user can't, like changing core system parameters (and even BIOS). On Linux, though, they run as your own user with regular privileges. This gives them access to your system only as much as any other app you are running. This is the key reason why some game developers don't let you play their online games on Linux, even when the anti-cheat system they use explicitly support it.

