The Current Situation

Normally, chess games end when there is a stalemate, resignation, draw acceptance, checkmate, or timeout. This is focusing on the ending of checkmate.

The Problem

When a player is put in checkmate, the game ends and the pieces are set back up for the next game, with the king being unscathed, which causes problems. The pieces are attacking the king with no way for the king to escape the check without going into another check, yet the king is not captured.

Why is this a Problem?

The game ends without a clear resolution; the king is trapped but not captured. It's like if a movie ended with the protagonist cornering the antagonist, but he's just like
"Ah well, looks like you got me; I forfeit," and the protagonist just leaves. It's just an unfinished ending; if the goal of chess is to defeat the enemy's king, then to defeat is to capture it, not to trap it.

The Solution

The simple solution here is that when in checkmate, the losing side gets one more move (it doesn't matter what it is since they're in checkmate), and then the other side captures their king. This does not affect the game when you are not in checkmate, because you still would not be able to move into check if it weren't already checkmate, meaning it doesn't affect stalemate or any other part of the game except for when checkmate is played. This would only extend the game by 1 move for each side, with the same result anyway (although the lost side could win if the other side ran out of time before the king capture). It makes the game feel more complete and resolute for both sides, making it feel more like a loss than a resignation.

To conclude this excellent petition, it's quite clear how it would be a wise decision to allow the capture of the king after checkmate in chess, making the game feel better and more finished while not affecting any other part of the game.