Am I to believe that cheaters would install Linux, just use a cheat in a game?

You seem to severely underestimate the extreme lengths cheaters will go to in order to cheat. Not only are modern cheats very expensive (like 20+ dollars per WEEK subscriptions), but the ones that are the hardest to detect require a second PC connected to the main PC using a direct memory access module so that the cheat can read the game's memory in a way that is impossible to notice for the Anti-Cheat running on the game PC. On top of that they spend time and money on stolen/farmed accounts, spoofing hardware and phone numbers, and buying entirely new PCs when they get detected and banned.

Installing Linux is a tiny obstacle compared to all the other shit these losers are willing to go through in order to cheat.