Interesting, why not?
New variation: simultaneous chess
I think this would be cool in that it would remove any advantage based on who went first. I did find this description of a version of simultaneous chess, but it still has something where the player with "initiative" makes their simultaneously chosen move first, and if the second player's move turns out to be an illegal move, the second player forfeits their turn and gets initiative for next time.
Instead, maybe there could be no initiative and instead have rules like if both players move to the same square, they both lose their piece, or something like that.

It will actually have more luck-based than regular chess
Regular chess is simply 1 person gets a move first while in this one, each move is mind games and will just eventually become a matter of who counters another with their RNG

This exists! https://www.kfchess.com/
Good luck...

This exists! https://www.kfchess.com/
Good luck...
That's fun! there's a cool-down for moving the same piece twice, and otherwise it's a clicks-per-minute exercise like many popular multiplayer games.
I thought the OP had in mind something more like the board game Diplomacy, where if two armies both try to move to the same spot on the board, neither one succeeds. That leads to situations where nobody makes progress... which is kind of the point in Diplomacy (the rules are designed so that the best way to break such a stalemate is to form alliances with other players), but it would be endlessly frustrating in chess. I can't think of a non-random way of breaking stalemates without resorting to "initiative" like cheesycow mentioned, which doesn't sound too different from taking turns whenever it actually matters.
What do you think about this variation idea:
The players move their pieces at the same time.
Once both players have chosen their move (as in pre-move), the moves are handled simultaneously. A piece is not captured if it moved away in the same turn.