Calling out to some chess-playing-function would be a deviation from the pure LLM paradigm. As a medium-level chess player I have walked through some of the LLM victories (ChatGPT 3-5-turbo-instruction); I find it is not very good at winning by mate - it misses several chances of forced mate. But forced mate is what chess engines are good at - can be calculated by exhaustive search of valid moves in a given board position.

So I'm arguing that it doesn't call out - it should gotten better advice if it did.

But I remain amazed that OP does not report any illegal moves made any of by LLMs. Assuming training material includes introductory texts of chess playing and a lot of chess games in textual notation (e.g. PGN) I would expect at least occasional illegal moves since the rules are defined in terms of board positions. And board positions are a non-trivial function of the set of moves made in a game. Does an LLM silently perform a transformation of the set of moves to a board position? Can LLMs, during training, read and understand board-position diagrams of chess books?