Realistically most Linux users are using a stock kernel and not something custom compiled. You can have both customization and a way to offer a secure environment for apps that need it. Even if you want to allow for custom kernels and drivers, the game could be setup to run in a secure virtual machine.
>The only way for this kind of anticheat to work is by introducing some part of the kernel that users can't touch.
To be clear, attestation is not anticheat. But yes, there would be components that end users would be unable to modify without removing their ability to attest to there being a secure environment for the game. Either these customizations need to be turned into policy for a trusted component to handle, or the customization needs to itself become trusted.
>but Linux isn't about obscuring the system from its owner.
Nothing about attestation requires obfuscation.
What you're asking for does exist though, in the form of Android devices and game consoles. Was curious about Steam Deck and... turns out it doesn't have secure boot. Someone could build a desktop OS on top of an anticheat-friendly kernel, but it'd probably not be big enough for gamemakers to care, and Linux desktop people would be uninterested in it to say the least. (I'm on a Mac btw, I have no horse in this race, just understand people who do)