A checkmate is not exactly the same as taking the king. I believe changing this rule could strongly alter the way we play. Because, if I am not mistaken, you cannot always capture the king after checkmate:
lichess.org/analysis/k6K/1rr5/8/8/8/1b3B2/8/8_b_-_-_0_1#2
Rh#7 Kxh7 and Rxh7 is illegal as the rook is pinned on b7 right? So this is a clear draw. Then if black is actually allowed to take the opponent king letting their own king in check, it is taken next move by the white bishop, so is it a victory because black did it first or a draw because at war both kings would be dead? Like in racing kings, you can have draws when you bring your king to the final square on the very next move after your opponent did it.
I don't know if this would give more interesting games to have such a variant and how to define good rules, but I find the topic is interesting.
lichess.org/analysis/k6K/1rr5/8/8/8/1b3B2/8/8_b_-_-_0_1#2
Rh#7 Kxh7 and Rxh7 is illegal as the rook is pinned on b7 right? So this is a clear draw. Then if black is actually allowed to take the opponent king letting their own king in check, it is taken next move by the white bishop, so is it a victory because black did it first or a draw because at war both kings would be dead? Like in racing kings, you can have draws when you bring your king to the final square on the very next move after your opponent did it.
I don't know if this would give more interesting games to have such a variant and how to define good rules, but I find the topic is interesting.