The foundational principle behind the rabbinic decree is stated plainly by the Rambam (Mishneh Torah, Hilchot Shofar 2:6): although blowing the shofar is itself only a rabbinic prohibition on Shabbat — and by strict logic the positive Torah commandment should override it — the Sages nonetheless enacted that we do not blow, out of concern that someone might take the shofar in hand and carry it through the public domain or from one domain to another, thereby incurring a transgression punishable by stoning; for while everyone is obligated in the shofar, not everyone knows how to blow it.
The Meiri (Rosh Hashanah 29b) makes clear that the act of blowing shofar is a skill, not a prohibited labor, and would be permitted on Shabbat by Torah law — yet the Sages decreed against it precisely because an unskilled person might carry the shofar to an expert in the public domain in order to learn, and Rav Mordechai Eliyahu (Maamar Mordechai, Chelek 3 55:55) notes that the same logic prompted parallel decrees regarding the lulav and the Megillah.
Sefer HaChinukh (405) underscores that the concern driving the decree is specifically for that other, less-informed Israelite — 'lest another Israelite take it and carry it four ells in the public domain and come to a prohibition punishable by stoning' — and the Kaf HaChaim (Orach Chayim 588:29) anchors this in Rabbah's formulation: 'All are obligated in the blowing of the shofar, yet not all are expert in its blowing.'
Rav Mordechai Eliyahu (Maamar Mordechai, Chelek 3 55:57) draws out the remarkable scope of communal responsibility embedded in this decree: the shofar is so exalted — likened to the innermost Temple service — and yet, because the Sages worried that even one person might forget and carry it four ells in the public domain, they suspended the mitzvah of shofar on Shabbat for all of Israel, for all generations, on account of that single individual.
Rav Eliezer Melamed (Peninei Halakhah, Days of Awe 4:9) restates the operative concern as it stands today: since everyone is obligated yet not everyone is versed in the prohibition of carrying in the public domain, those who do not know how to blow might take their shofar and go to an expert to be taught, and in so doing would transgress the serious Shabbat prohibition — and the Shulchan Arukh (Orach Chayim 588:5) codifies the result: when Rosh Hashanah falls on Shabbat, we do not blow the shofar.
The dimension of responsibility for every individual Jew — not only the learned majority — is illuminated from a different angle by the Alshekh (Vayikra 21:1), who teaches that a leader cannot rest content that 'whoever hears will hear and whoever ignores will ignore,' for God instructs Moshe to return and warn even the single individual who needs repeated attention; and the Minchas Yitzchak (2:9) likewise reflects Rabbah's ruling that the shofar is not blown when Rosh Hashanah falls on Shabbat because of the decree lest one take it in hand and carry it to another who will blow, framing the entire enactment as an expression of that collective obligation.