Why can't anti-cheat software work on Linux?

Valorant and the like use kernel modules. They run along side the most critical portions of the OS, just like many hardware drivers. This allows them to detect anything happening on the system.

Kernel modules need to be written for the operating system, so this means there would need to be a userbase present on Linux/macOS to justify writing such a driver (writing kext for macOS is even more difficult as Apple has more or less banned them).

You don't want these types of anti-cheat. I would encourage anyone looking into games that use kernel modules for anti-cheat to avoid them.

Unfortunately user-mode anti-cheat has been defeated time and time again.

You're mistaken, anti-cheat (some of them) do work on Linux[1], such as Easy AntiCheat, BattlEye, nProtect GameGuard etc. But not every game using these will work, since some of them require the developer to enable Linux support - which is usually as simple as ticking a box. Unfortunately, many game studios / publishers don't want to support Linux, for various reasons. Some of them even are even biased against Linux, such as Epic's Tim Sweeny, who has made several unsubstantiated nonsensical claims against Linux. Unfortunately there's not much you can do about these people / studios, besides boycotting their products and services.

There's also a second category of anti-cheat (kernel-level), which will never work on Linux, as @p_ing elaborated. I second their opinion that you shouldn't run games which use these, even on Windows. The Crowdstrike incident has shown us how dangerous such programs can be (causing BSODs and instability). Kernel-level anti-cheat programs are even more dangerous, since they monitor everything on your PC, actively sabotages any inspection tools, and protects itself from being uninstalled - which is behaviour very akin to malware/rootkits. You should _never_ install such programs on your PC. Once again, the solution here is to vote with your wallet and boycott games and studios which use such solutions.

[1] https://areweanticheatyet.com/

Technically, anti-cheat requires control over the system (computer being used, room and network where the test happens).

An open source operating system could be used to block or circumvent the control check points of said spyware. So installing it on your Linux workstation may not provide enough guarantees.

But... the unasked question, but important one is: why install one on _your_ computer? If someone (school, org, whatever) needs you to pass a test, they can provide the whole package if they want to exert control over the whole context of the test.

But submitting in advance to their own requirement (it's not _your_ requirement), by compromising your own computer?

This may as well be a hidden purpose test, where refusing to install said software is the "good" answer. And if the employer doesn't accept/understand that, that's... problematic for them on a whole other level.

Be psyched that it doesn't. Anti-Cheat software is spyware at it's finest.

There's more variation. Every Windows system is basically alike. Anything unusual is a cheat. Every Linux system is not alike.

You could make one that works for Ubuntu.

Anti-cheat often tries to detect suspicious modules, for example. On Linux, that suspicious module could be a buggy or obscure driver for some obscure piece of hardware. On Windows, at least it's signed by Microsoft saying "this is a real driver for an obscure piece of hardware".

Namespaces allow to fake filesystems, processes, users, devices. selinux, apparmor, ... can deny access for the anticheat.