It's a complex aspect that needs consideration. The only reason the king isn't captured in chess is because it's unecessary, if a king is in checkmate he can be captured next go no matter what. That isn't the case here, so it's a drift away from classical rules.
Then again, if we allow Ki15, awaiting green's move, that's also a drift from classical rules.
How does chess.com standard 4p deal with this?
4 player chess at chess.com does it the same way as this game (in the teams variant). Kings are not allowed to be left in check and it is considered mate at the start of that players turn. You are correct though that a checkmate does not necessarily mean the king will get captured no matter what.
In free for all you can be checkmated after each players move (not waiting until it is your turn before it is considered a checkmate).
I've played one 4p team game and quickly got checkmated, I can't say I enjoyed that game.
As for free-for-all, well I seemed to do better here, simply because I was conservative, sitting back and watching the others trade pieces and push pawns. But I didn't really like the fact that the result is not in my own hands. Still, it was fun enough for me to play a few games.
I know "collusion" is not allowed in 4p chess. However, if you see someone is in check, well you can attack his queen, can't you? Is that collusion? Or is that playing the best move?
I've played quite a lot of 4p chess. I have managed to get into the top 10 highest rated ffa players once and I am an admin in the official 4 player chess club.
I think both variants can be very fun. It is easy to get checkmated early in the teams variant, but that is because it is a very tactical variant. Games between 2 strong teams can be very exciting.
In ffa collusion is indeed not allowed, but taking advantage of weaknesses, such as winning a queen against a player that is in check is perfectly ok. It just means you need to be very careful about putting yourself in check to the player that plays after you (since there's 2 players after him that can take advantage of the check before it is your turn.)
This is checkmate, yes. It is yellows turn, yellows king is threatened by an enemy piece and he has no legal move to avoid being in check.
If we were playing a variant where the goal is to capture the king, then yes there would be more moves to intervene, but then you have to allow players to leave their king in check.