Why isn't chess an impartial game?
Chess violates all three conditions, so I don't really understand what there is to ask.

(1) The game is not finite. Although the 50-move and threefold repetition rules allow a player to end the game under certain circumstances, they do not oblige the player to do so.

(2) The game has draws.

(3) As you have observed, the game definition distinguishes between the players. Your "active player" trick doesn't work because you've turned chess into a one-player game: the "active" player makes every move and the "inactive" player is a spectator.