I do gaming livestreams from a Linux setup. Out of the last dozen of games that I played, most worked perfectly right out the box. One had a minor texture glitch on the start screen (though not in-game), one had rather bad (though playable) performance. The only issue that I've seen across multiple games is that they seem to get confused when screens are on different resolutions, but I can usually fix that by flipping between windowed and fullscreen.

So yeah, it's not all sunshine and butterflies, but "extremely low number of games that work out of the box" is not true in my experience.

(Important side-note: I don't play online multiplayer stuff, so the whole anti-cheat software topic does not apply to me.)