Kinda cool, but wouldn't it always favor white?
3-move chess

Hard to say, I think it would be equal, if not more fair than traditional chess; given the wider range of openings. What would be your most favorable 3-move opening for white? Careful, don't hang your pieces.

How would checkmate occur, If it would be capturing king=win then white will always win. Also if a piece attacks another piece it could be taken. The game would be so boring! The number of opens is almost zero... if you move your queen anywhere that can be attacked in one or two moves then it is bad since after 2 moves to attack your queen you take it! You can take any piece then move back. Ex. white attacks a black knight, then he takes it, and he moves back. In general this mode will not work

I like the idea, I think a two move chess would be possible, definitely not ranked but for fun. You'd have to anticipate moves and defend more positions

I think the comments kinda showcased some of the flaws already 8)
You're welcome to try a game here in this forum though.

This is a very similar idea to one already suggested, called Double Chess.
https://www.chess.com/clubs/forum/view/marseillais-chess-double-chess
Chess but each player makes three moves at a time.