Neural Nets can be trained to play chess very well and have been doing so for a long time (see Stockfish and Leela as some of the most popular/strongest ones - top GMs have no chance against them), but these are dedicated models, where the game rules are encoded in the learning process, as opposed to large language models which are natural language processing models. Technically you can give an LLM a lot of chess books and games and it will be able to spit out chess notation. Put a webapp on top that renders text moves to the board and it looks like it's playing chess. But it isn't really.