You should not consider Tim Sweeney's comments on the matter as a reliable source. He was veiling his true motivations behind that statement. The Switch does not run Linux either, it's a custom OS descending from the Wii's iOS.

The cheating issue isn't really a matter of being able to run custom kernel code. You can do the same thing on Windows, which is why remote attestation is a thing for some games. As someone who has developed games for Linux (and Windows / Mac), it's an endless cat and mouse game. So long as the system can execute code that is not yours, you never really are getting perfect anticheat. Ease of loading custom kernel code isn't really a hurdle to that.

I find that client and server based in combination is the robust approach. I once implemented anti-cheat in which the server lied about game state, which a regular client without cheats would act predictably on. Deviation from that behavior is a useful heuristic to build a suspicion score.