You could make a Linux distro with a signed boot chain and a kernel anti-cheat, then you'd mostly need to get developers on board with trusting that solution. Nobody is doing that today, even Valve.
Funny enough, macOS of all things is maybe "best" theoretical platform for all this because it does not require you to trust anyone beyond Apple. All major macOS programs are signed by their developers, so macOS as an OS knows exactly where each program came from. macOS can also attest that it is running in secure mode, and it can run a process at user-mode level such that it can't be interfered with by another process. So you could enforce a policy like this: if Battlefield6.app is launched, it cannot be examined by any other process, but likewise it may run in a full sandbox. Next, Battlefield6.app needs to login online, so it can ask macOS to provide an attestation saying it is running on genuine Apple hardware in secure mode, and then it could submit that attestation to EA which can validate it as genuine. Then the program launch is trusted. This setup requires you to only trust Apple security and that macOS is functioning correctly, not EA or whatever nor does it require actual anti-cheat mechanisms.