Look at what happens at the end of an ordinary rook mate. The rook lands on f8, the black king on h8 is attacked, and the players shake hands with every piece still standing exactly where it was. The black king is right there on the board.
8 
f Yes, White could take the king next move. That move is never played, because the game is already over.
The rules stop one move early, on purpose
Article 1 of the FIDE Laws sets the objective of chess as putting the enemy king under attack in such a way that the opponent has no legal move. Not capturing it. Attacking it inescapably. The moment that position appears on the board the game ends, and anything either player does afterwards is not part of the game.
That is not a technicality invented to be tidy. It follows from what check is. A check has to be answered on the spot, so a position where the check cannot be answered is already decided, and playing out the capture would tell you nothing you did not know.
You are not allowed to expose your own king either
The other half of the rule is the one that does the real work. No move is legal if it leaves your own king attacked, which means a player can never volunteer the king for capture, even by mistake, even deliberately.
d e That is why a pinned knight cannot move, why you cannot castle through an attacked square, and why a king can never take a king: the square next to a king is always attacked by it, so the capture is illegal before it is anything else. The idea that you take the king anyway survives because plenty of beginners play their first games exactly like that, grabbing it when the opponent misses a check, and nothing bad comes of it: the winner would have been the same either way.
The habit worth building instead is counting. In front of every mate is a check, and in front of every check is a king with a shrinking number of squares. Mate in one puzzles are nothing but that count, done a few hundred times until it is automatic.