I'm not sure COT would help in this situation. I am an amateur at chess but in my experience a large part of playing is intuition and making and I'm not confident the model could even accurately summarise its thinking. There are tasks in which models perform worse on when explaining reasoning. However, this is completely vibes based.

Given my experience with the models, giving it the ability to think would allow it to attend to different ramifications of the current board layout. I would expect a non trivial performance gain.