Earlier quoted context omitted.
Fun idea, but what happens when a player submits an illegal move that becomes legal only when taking the opponents move into account or that’s only legal before the opponent’s move? Say a rook moving from a1 to a3 while an opponent’s knight moves from or into a2? Also, if you are going to write down moves, you can allow the players to move all of their pieces, too, like in Diplomacy. A queen could capture a pawn next…
I think I would specify that a move has to be legal when at the start of the move, i.e. while it is being written down.
There also is the possibility of simultaneous checkmate.
Another edge case is pawns taking pieces. Could a pawn move diagonally into a square being vacated by the piece standing there? Could two pawns simultaneously not take out each other (say by the simultaneous moves d4×e5 and e5×d4)?