Why isn't chess an impartial game?

Question

In Combinatorial Game Theory, a major distinction is drawn between impartial games and partisan games. To be impartial, a game must satisfy these conditions:

(1) The game is finite; i.e. there is a constant $c$ such that all games end in $c$ moves or fewer.

(2) There are no draws; either player 1 or player 2 wins.

(3) The rules of the game draw no distinction between the players. That is, given any position, if we switch the active player then the set of legal moves is still the same, and the player who wins the game under perfect play switches.

It is often said in introductory CGT resources (here for example) that a major class of games that violate condition (3) are those in which each player has their own piece set. For example, chess violates (3) because the white player can only move the white pieces and the black player can only move the black pieces, so if we switch the active player then the set of legal moves changes.

However, it seems to me that this problem can be circumvented by the following trick. Instead of defining a "position" as the locations of the white and black pieces, we define a position as the locations of the active player's pieces and the passive player's pieces. Now the conditions in (3) are satisfied.

This trick seems too simple to be original, though, and I have read in many places about how chess is a classic example of a partisan game. So what am I missing?

Aside: To circumvent (1) and (2), we need to modify chess in some reasonable way; i.e. forbid 3x repetition of a single position and declare stalemate to be a win for the stalemating player. But that's beside the point of my question.

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Years-later edit since this question has gotten some random attention lately: neither of the answers really understood the point of the question (which is probably my fault for an unclear OP). My current understanding is that there is indeed a natural isomorphism between finite drawless chess and an impartial game, but impartiality is not a property that is always preserved under isomorphism, and my misunderstanding came from the assumption that it was.