Why do we blow Shofar on Rosh Hashana and shake the Lulav on Sukkos? I would say that a Shofar lasts longer, so it would make sense to use it on a longer holiday, while the Aravos wilt within a day or two, so use them on a short Yom Tov like Rosh Hashana instead?
Because if we'd blow the Shofar on Sukkos, we'd have to use it for Hakafos, and the last time we did that (circled something once for seven days, and then seven times on the last day, all the while blowing the Shofar), we managed to bring down the city walls.
And since rebuilding the Bima every year is kind of expensive, we do it with the Lulav instead.
Because if you blow the lulav on Rosh HaShana, it's liable to split the תיומת.
And shaking the shofar on Sukkos wouldn't make any noise; unless you put something in it, in which case you"re making your mitzvos into chavilos.
Ideally, we would have everyone take a shofar for himself on Succas. Each man shall take for himself a beautiful fruit of the ram, rejoicing before the Lord for seven days. Of course, the shofar couldn’t be bent past m’lo kumtzo, and a spot bigger than k’gris would make it pasul altogether.
However, we have backed away from this practice based on the Talmudic discussion (Rosh Hashana 30a), which describes the practical noise issues which arise when may people begin to blow at once.
מאי אין כל יחיד ויחיד חייב לתקוע אילימא דביובל תוקעין יחידין ובראש השנה אין תוקעין יחידין והא כי אתא רב יצחק בר יוסף אמר כי הוה מסיים שליחא דציבורא תקיעתא ביבנה לא שמע איניש קל אוניה מקל תקועיא [דיחידאי]
The Gemara asks: What is the meaning of the clause: Each and every individual is not obligated to sound it? If we say that on Yom Kippur of the Jubilee Year individuals sound the shofar, whereas on Rosh HaShana individuals do not sound it at all, this is difficult: But when Rav Yitzḥak bar Yosef came from Eretz Yisrael to Babylonia, he said: When the prayer leader completed the sounding of the shofar in Yavne, nobody could hear the sound of his own voice in his ears due to the noise of the sounding of individuals. This indicates that individuals would sound the shofar even on Rosh HaShana.
You have some good points (pun intended, BTW). There are some pretty simple answers, though.
1 - The simplest reason is that Rosh Hashanna is called a Yom Tru'ah - a day of "blasting". Well, you can see some of the practical problems of blowing a lulav:
- You'd get a lot of sharp "thorns" in your mouth and tongue
- There are supposed to be 3 "distinct" types of sound. How would you distinguish between tekiah, shevarim, and teru'ah by waving a lulav? To me when you wave a lulav it sounds like leaves rustling in the wind. That's it.
2 - This one is just more of a question of shul "cleanliness". I find that regardless of how clean the lulav is, there's always some lulav "hay" that falls off. Now, you have that hay all over the seats for a long davening. Even if you have Gentile maintenance who clean up the shul, they can't clean it up until after services are over. I know that I don't want to sit in messy hay for a few hours.