> Isn't that how LLM models are trained right now

It's neither how computer chess works or how LLMs are trained.

Computer chess uses various tricks to prune the search space of board states, where the search is guided by the "value" of each board state. Neural networks can be used (and probably was at the time) to approximate this value, but there can be hand coded algorithms with learned statistics or even lookup tables for smaller games than chess.

There's no search in LLM training.