> you'd quickly find that it isn't able to track a series of moves for very long (usually 5-10 turns; the longest I've seen it last was 18)

In chess, previous moves are irrelevant, and LLM aren't good with filtering out irrelevant data [1]. For better performance, you should include only the relevant data in the context window: the current state of then board.

[1] https://news.ycombinator.com/item?id=44724238