That's very much against the philosophy of Linux. At best you'd divide Linux gaming into trusted (known operating system, hypervisor, no root access), and untrusted systems.
It's essentially what Google are trying to do with attestation, Web Environment Integrity, etc.
Edit: there's no way to stop cheaters without also stopping software freedom in general. The best path forward might be to focus on building communities of people who enjoy playing games together.
Who's going to pay to develop it? Who's going to support it? Who's going to patch it when there's new kernel versions? You know how fucking complicated kernel programming is?
From the top of this thread, Valve was suggested as a candidate for someone who might already be interested in these things, perhaps to the point of invested into each of those.
Or maybe they don't. Maybe nobody does.
People can speculate and dream. Nobody's speaking authoritatively here, and certainly nobody is petitioning that Linus himself get down and dirty in anti-cheat functionality.