On Daf 43 of Succos we discussed some of the factors behind the strange practice of not blowing shofar on Rosh Hashana, and that the official reason of lest one carry seems to be only a surface level explanation.
Further in support that the official reason of fear of carrying is not sufficient, The Yismach Moshe (Emor 14) cites our Gemara where it is suggested that Bais Shammai could hold that carrying in a public domain is forbidden on Yom Tov just like on shabbos. If we were to hold by the reason for not using a shofar on shabbos Rosh Hashana is lest he come to carry, how would Beis Shammai in this section of the Gemara ever allow the use a shofar, since carrying on Yom Tov is also forbidden!
What then is really the reason? Yismach Moshe says that we cannot know the true purpose of shofar. All the hints such as it is to remind us of the Akeidah or to wake us up to do teshuva are actually potential misappropriation of God’s mitzvah. Imagine, he says, if we used a king’s trumpet for our own purposes? In truth, we would have no permission to blow the Shofar at all. However, because Yom Tov has a special heter of Ochel Nefesh, that is we are allowed to do work on Yom Tov for our basic needs, this is symbolic of the entire spiritual zone of Yom Tov. It is a space where we can take the spiritual and use it for our physical needs. Thus, the Shofar is permitted on Yom Tov because we are given a special license to appropriate spiritual matters for our own use, but not on Shabbos.
Translations Courtesy of Sefaria