I am sure this idea has popped up many times, also in combination with fairy pieces.
Main problem is that to eliminate all opponent pieces you will also have to eliminate his Queen. And you need to have an unrealistically large advantage to be able to 'checkmate' a Queen, even if it is the opponent's only piece. So by avoiding to trade your Queen you basically cannot lose. And since both players know that, the game will always end in a draw. Not very interesting.
Even if the Queens are traded, the same thing pretty much applies to Rooks.
You want to take all other people’s pieces to win.
Please tell me if it is already a variant.