Why shouldn't a person in their year of avelus be a shaliach tzibbur on shabbos or yom tov or yomim noraim?
This is indeed the custom as recorded by the Rema in SA YD 376:4
The mourners recite Kaddish even on the Sabbath and Festivals, but it is not customary for them to lead the services on the Sabbath and Festivals, although there exists no prohibition against [this] matter.
Commentators note the conflict between the elevated level of joy in the prayers and the internal state of the mourner. For instance, regarding Mincha of Shabbat, the commentary Daa's Torah writes that the mourner doesn't recite the prayers since they include the words "Avraham exults, Isaac rejoices etc." and "Cause us to rejoice in your salvation". Similarly regarding Yom Tov, he writes a mourner may not lead the service because they include "jubilant rejoicing".
During Rosh Hashana and Yom Kippur, a mourner may lead the service if there is no one better qualified (Schach YD 376:14).
Source: R Chaim Binyamin Goldberg's mourning in halachah, R David Brofsky's Hilkhot Avelut