Several years ago I systematically searched the quotations within the OED for instances of singular subject with "their" in the subject. I hope no one will be shocked that there were many many examples going back hundreds of years. Such a construction seems superior in some ways to the suggested alternatives:

"Players must move their piece..." seems to imply that some number of players have joint possession of a piece.

"Players must move their pieces..." might be seen as implying players jointly possessing multiple pieces.

"Each player most move their piece..." violates a formal rule but conforms to historical (if not very recent) usage and indicates each player has possesion of a piece.

I see I am agreeing with and (I hope) reinforcing the point of "Genesius."