Chess is a war game, so allowing political assassinations or allowing the King to die just doesn't make any sense. Assassination of the King would just mean that the next in line becomes the new King. Only the King can surrender. So in order to force an end to the war you need to trap the King. Killing the King does not end the war, it just creates a new King.

I mean, that's kinda how it works when playing 1 on 1 with no audience and one player misses moving into check, and the opponent misses too.

And there's plenty of people that play with no check at all. You don't have to announce it, it's on the player to detect and counter.

I prefer check being absent in casual, friendly games unless they're so friendly that you're also notifying the other player of missed opportunities or mistakes. Which is, imo, the optimum way to play with friends that aren't at the same skill level, or are very closely matched to the point that the game is going to drag on when there's limited time to play.

What's stopping you from doing that anyways? If your opponent doesn't see it, they don't see it

A lot of chess variants do this, like Duck Chess and Drawback Chess. It's especially useful if there's a possibility of something outside the core chess rules (e.g. the duck or a player's drawback) that could actually stop them from taking the king.

One extra detail to be aware of if you want to play this way: you should (or should at least consider) add in "castling en passant", where a piece landing on the space a king left on the turn after it castles, or on the space the king passed through in the act of castling, also counts as capturing the king.

Your ideas a fun to think about but I don't see them as improvements