LLMs can be trained on chess games, but the tree of possible board states branches so fast that for any given position there is simply very little training data available. Even the billions of games played on chess.com and lichess are only a drop in the bucket compared to how many possible board states there are. This would have to be split further by rating range, so the amount of games for any given rating range would be even lower.
This means that the LLM does not actually have a lot of training data available to learn how a 1300 would play, and subsequently does a poor job at imitating it. There is a bunch of papers available online if you want more info.
LLMs already do play at elo ~1400-1800. The question was how does their style feels like to someone who can appreciate the difference between a human player and a chess engine (and the different styles of different human players).
This means that the LLM does not actually have a lot of training data available to learn how a 1300 would play, and subsequently does a poor job at imitating it. There is a bunch of papers available online if you want more info.