Good!
Real Time Chess

This is a great idea! I especially like Frame 4, as it shows how this variant would force players to think outside the box. New tactics unfold with pawns being able to move in different ways. +1

So, would black win all captures where pieces move to the same square? And the variant would have to allow moving into check and king captures as well.

So, would black win all captures where pieces move to the same square? And the variant would have to allow moving into check and king captures as well.
You are right, some things do need to be ironed out as it were like: what happens if a piece tries to block a check for instance? What if two pieces capture each other at the same time? What id two pieces land on the same square at the same time?
By all means, give us some suggestions! I'm gonna try and work out a lot of the perticulars at a later date but this should be enough of a start.. the main idea of players going at the same time is there after all. How do you think we should handles checks?

What if the white made a move that was illegal? Like moving a pawn where black's pawn is? Does white get the capture?

Ayy, ty.
I wrote in more detail about this topic here if it interests you:
https://www.chess.com/blog/HighEldar/time-frame-strategy-gaming
UPDATE: I later wrote a blog about this topic here: https://www.chess.com/blog/HighEldar/time-frame-strategy-gaming
I still don't know what to call it between Real Time Chess, Time Frame Strategy or At-the-Same-Time Chess....
I've got it. Biggest idea for a Chess variant in a century.
Real Time Chess.
Each player makes their move "at the same time". By this I mean, each player makes a move within the same alloted "time frame".
For example, rather than White going first then Black, both players make a move at the same time... when one player makes a move on the board, this move is not seen by the opponent. When the second player makes their move, both moves are now visable on the board and take effect.
Clocks
Regular: When one player has locked in their move, the other players clock starts counting down.
Delayed: All players are given same delay to make their move and the clocks start counting down when a move takes longer than this. The whole board is updated only once all moves have been made.
Example
Starting Board:
After 1 "frame"
After 2 "frames"
After 3 "frames"
After 4 "frames"
In Frame 4, white attempts to capture the black Knight but the black Knight was able to jump out of the way so the pawn moving to that square is a legal move but the capture is not legal because black jumped out of the way.
Real time chess can be played with virtually any board variant including fiary game versions, 4pc, 8pc and much more.