One of the most painful moments in the light novel... A conversation about chess that makes very little sense. It hasn't started making more sense once adapted.
It begins with Elliott thoughtfully declaring the game to be a "stalemate", with a tiny view of the position. The definition of "stalemate" is a narrow one - it happens when a player is out of legal moves, but also not in check. So far as we can tell, only White could possibly be in stalemate. Assuming the black pawn is cutting off two of the squares to the king's sides, and the white pawn diagonally adjacent is being pinned by a bishop, and the tall piece cut off by the top right edge of the panel is a king, and the black pawn in front of the king is protected somehow, the position could indeed be stalemate, though an incredibly contrived one. Black was definitely winning massively and blundered something, which doesn't mesh with the game being later described as Monica fighting Elliott to a stalemate on equal footing.
An adaptation prioritizing chess accuracy (the only adaptation worth doing) could do so by Elliott proclaiming a draw instead of a stalemate, which is infinitely more flexible to work with, and a much more common outcome of a chess game. A natural one for two players meant to be on equal footing at this point in time.
Elliott proceeds to philosophize about the interplay between chess and personality. Cyril being described as a "protect-the-king" type of player is somewhat plausible, if translated into a player who prioritizes king safety in the opening and forgoes early aggression; say, opening Nf3 - g3 - Bg2 with White and castling and deciding on a course of action afterwards based on Black's response. Monica, meanwhile, is characterized as a player who would gladly use their king as "bait" in order to win. That's not really a thing. There is such a thing as aggressive, risk-taking chess, of course, but the risk being taken is usally sacrifices of pawns or pieces to expose the other player's king and/or get a lead in development, hoping to checkmate before the material disadvantage proves decisive. I can't really think of any gambit opening that specifically sacrifices king safety - that's a good way to just lose. What's true is that a player who has the initiative (they are delivering threats move after move the opponent must parry or else) may not have to castle or worry about king safety for a while.
Up to this point, there's a possibility the author had shogi in mind while writing. By my very limited understanding, one of the big differences between shogi and chess is that castling in shogi takes multiple moves, and players may choose to castle very thoroughly (spending a lot of moves, and leaving the opponent to their own devices for a long time) or form a quicker castle that is easier to attack, or just leave the king in the middle and go on the hunt right away. That absolutely could be characterized as using the king as bait. In chess, it could be adjusted into Monica being a player who routinely sacrifices pawns to attack early and often, as opposed to Cyril, who holes up with his king and hopes to win in the long run.
The Jean-Luc Picard facepalm moment is Elliott's declaration that he refuses to promote pawns on principle. It makes him sound like a classist douchebag, but worse still, it makes him a terrible chess player. How is he ever going to win an endgame? And this part wouldn't make sense in shogi either. Promotion is an even bigger part of shogi than it is in chess, if anything!
Overall: the chess talk in this chapter could work for the most part, with some adjustments. It's just Elliott who has to think of a way to be a classist prick without making taking a nonsensical stance against fundamental rules of chess. I honestly can't think of one myself, unfortunately. The problem is that Elliott is meant to be genuinely convinced that a strict division of the social strata protects commoners as well, which means his chess style can't be one that gleefully sacrifices pawns. I think chess just wasn't the way to showcase this aspect of Elliott's character (assuming it's even worth showcasing!)