And unimaginably long list of moves, yes. But literally infinite? No.
Wrong, I'm afraid. A game of chess could be infinitely long. Not really sure why you think it couldn't be.
Because of the three move repeat rule. There’s no conceivable way the game doesn’t eventually end. Even if players are avoiding checkmate there’s a finite amount of positions (albeit an extraordinarily large amount), and eventual you have to repeat, checkmate, or choose a position that is drawn by rule.
You're still wrong. You claimed that a chess game could not have an infinitely long list of moves. But it can. The three-fold repetition rule does not mean the game is a draw; it just means that a player can claim it as a draw. If neither claims it as a draw, the game continues, ad infinitum if so.
So, once again, a chess game can involve an infinitely long number of moves. This is just a simple statement of truth.
From the theoretical point of view, this is of no importance. All that matters is that there are a finite number of legal positions. When each of these has a precise value (as in a tablebase) the much greater number of possible legal games can be perfectly analysed, and every mistake precisely identified.
You didn’t explain anything.
How does an engine beat a person without the person making a mistake?
If you have no good answer, please indicate so by deflecting with something like, “I already answered that.”