What is the nature of the game of chess, and what makes a game a chess variant? Here is my opinion:
"Chess", in the widest sense (ie. in the sense of a family of games), has certain properties:
1. It is played on a "board" with "pieces". Each piece occupies a position on the board, and each position on the board can be either occupied by a single piece, or can be empty.
2. It is a competitive game (ie. a game where a player attempts to win and tries to avoid losing) played between two or more players.
3. The players alternate making moves in a "round-robin" sequence. A move is the changing of position or type of one or more pieces on (or off) the board.
4. When a piece is moved on the board, that movement may result in a "capture", which is the removal from play (ie. from the board) of one or more pieces.
5. There are at least two different piece colors. The color of a piece determines the rules for which player may move that piece.
6. There are at least two different piece types. The type of a piece determines the rules for how that piece may change position on the board.
7. A piece may also be brought back into play (ie. moved from off the board, back onto the board) or may change type while on the board.
8. It is a game that involves no "hidden" information (except for the thoughts of the opponent(s)). All events that occur on the game board are determined by rules and information that are known prior to the start of the game, and known by all players.
9. It is a game that involves two classes of pieces, "royal" and "non-royal". The winning condition of the game is to capture or checkmate one or more of the royal pieces.
It is interesting to note, that the game of checkers, has the same properties, except for the last. I believe it is this last property that differentiates chess from other non-chess games (such as checkers).
If you change the winning condition of, let's say FIDE chess, such that the winner is the player that captures all of the opponent's pieces, then, in my opinion, that game becomes much more like checkers than chess. Similarly, if you changed the winning condition to be the player who is first able to move their King to the last rank, I think the game will play more like parchesi, or one of the other "race" games.
That is not to say that one cannot change the winning condition and claim a new chess variant. Certainly in the case of Extinction Chess, that has been done. And certainly any of the properties above could be changed to produce a new chess variant. It just seems to me that the last property is the most crucial to making chess unique from other board games.