Anti cheats basically don't work on Linux at all. One of the primary jobs of anti cheat software is to monitor the OS environment and detect people trying to inject code into the game process or read the game process's memory. On Windows kernel access is gate kept by Microsoft and restricts cheat software's options for how to read memory and inject code. The standard Win32 APIs are monitored and the cheats need exploitable kernel drivers to get in.
On Linux this can't work because a cheater can just build their own kernel with all the protections disabled or with intentional vulnerabilities. From what I've heard, statistics for games running anti cheats on Linux alongside Windows find the vast majority of cheaters on Linux.
Given this situation I think it's entirely reasonable to not support Linux if you're handing cheaters the game on a silver platter.
Assuming the Anti-Cheat won't be GPL licensed (which would probably make it useless), the Linux kernel does indeed refuse to cooperate: https://lwn.net/Articles/939842/
Sure, there are technical solution around this, but they are legally questionable.
Linux doesn't refuse anything, it's free and open source software. If publishers want to offer anti-cheat software developed for Linux, it will run. In fact, many games do have anti-cheat, like Insurgency: Sandstorm, which uses EAC through Proton.
Publishers can even develop invasive kernel-level anti-cheat just like they do for Windows. They don't because it's a small portion of the market currently, and I assume they consider it not worth the investment as of now. To what extent existing Linux users would willingly allow such software to run is also an open question.
There's also the point that even invasive kernel level anti-cheat on Windows with requirements for secure boot continues to be inadequate to stop cheaters in competitive online games.