Is a Bar Mitzvah Boy an Adult?

The boy spent weeks in preparation for his bar mitzvah. When the moment arrived, he relished the opportunity to read the Torah. He had finally achieved adulthood. He was now a full card-carrying member of the tzibbur. Or was he?
Our sages have taught that there are physical attributes that signify when a boy has become an adult. However, there is an alternative to a physical inspection. We can rely on a chazakah. We can assume that when a boy turns 13, he possesses the physical attributes that render him an adult. Practically, everyone nowadays just relies on this chazakah. However, the Mishnah Berurah writes that this chazakah should be used only for mitzvos d’Rabbanan. We will allow a 13-year-old boy to daven at the amud and lein the Torah, but he should not blow the shofar for a congregation. Tekias shofar is a biblical mitzvah for which we don’t rely on this chazakah.
The mitzvah of Kiddush on Friday night may be a mitzvah d’Oraisa so preferably an older adult should not hear Kiddush from a boy who is merely 13. (Some suggest that after davening, the mitzvah of Kiddush on Friday night may be only rabbinic.) Likewise, a bar mitzvah boy should not write the parashiyos of tefillin if his adulthood is established on the basis of the chazakah.