Fun little game!
I would agree that the punishment of falling should be enough for the player as they are losing progress. Killing them from fall damage doesn't really make sense in this type of game. Think of jump king when you fall you are already losing. You don't need that extra kick of losing completely because say you try for 10 minutes to get up an area you just lost that 10 minutes.
Like others said the platforms that fall apart are a bit misplaced in this type of game. It works if the player dies but otherwise not really as there is no way to progress again if you mess up.
Great job!