There is no direct mitzvah to blow the shofar on Rosh Hashanna. The mitzvah is to hear the shofar blown. It seems that this mitzvah is phrased "causativly". In order to hear the sound, someone has to blow it! Yet, the blower doesn't get any direct mitzvah for the act of blowing; he "passively" gets the mitzvah when he hears his own blowing.
Why is there no mitzvah for actively doing the blowing itself? As for those that cannot blow, technically, they could fulfill the obligation by someone else doing it, no? Wouldn't that work the same way as someone making Kiddush for another?