Since there is checkmate in 3-checks chess, there is check also. So for check to be meaningful, it has to mean that the side that is in check, has to move his king out of check in the end of his turn.
So yes it's perfectly logical that you can't win by giving a 3rd check as you have to make a move that will remove the check from your king.

What you say would be logical if there was no check and one in order to win will have to either capture the king(and not checkmated him) OR threaten him(check him, but the term is ambiguous in this context) 3 times. Whoever manages to do one of the 2 first wins.
Then your 3rd threatening("check") on the king would win even if at the same time the opponent was threatening to capture yours.

It's like atomic Chess where even if your king is threatened exploding the king has a preference, so if you can leave your king hanging by delivering an explosion on the opponent's king, you win. But in atomic there is no checkmate.

What you propose is a different type of game with different strategy.