Perhaps the solution here is not to do personal/sensitive work on gaming machines, but then we're right back to having separate game consoles...
Perhaps an ideal scenario would let a player choose whether or not to enable a game's anti-cheat, but that it'd be required for certain matchmaking and (obviously) competitive features. So that I can opt out if I'm just trying the game casually or just playing with friends in a private room, but can enable it if it's something I intend to play seriously and am invested in enough to give deep access to my system?
The main weakness (other than code vulnerabilities) is the network: it's very difficult to guarantee that a remote computer is running the allowed OS & game code, and not a hacked version. However, technologies like Intel's SGX x86 extensions can make this possible: because of the added encryption in the core CPU, one machine can send a cryptographic challenge to another one, and it can only respond correctly if the remote CPU has validated the code it is executing.
So, the tech is there. But in reality, people really don't want to lock down their PCs so securely and effectively give full control to their OS vendor.
But, it could upload "suspicious" data to the home servers to be evaluated and fed into the signature database. Just like any other feedback loop in anti-cheat (and anti-virus) systems, only this one can access everything on your system.