That's what a human will do if they already have a physical board to look at. But if someone, say, posed you a chess exam question via FEN notation, or as a sequence of moves in algebraic notation, you'd sketch a visual representation of the board off your own initiative to help you answer the question. There is nothing in principle to stop the LLM creating its own board representations in whatever format enables it to easily keep track of game state and legal and illegal moves. If it fails to do so, that's a sign of its own limited understanding of chess as compared to a human.

The LLM would only be playing 'blindfolded' if you somehow forbade it from making notes (as you effectively do by literally blindfolding a human, given how limited human working memory is). But you are not doing that. The LLM is free to keep track of the game state via whatever means it chooses.

None of this is about superhuman ability. Any human who understands a given chess notation can convert it to a visual representation of a chess board and then use that representation to choose their next move, with their usual level of performance.