According to this question, the Annotated Tehillat Hashem Siddur (a Chabad siddur) encourages the Bar mitzvah boy to get his first aliya on Monday, Thursday, or Shabbat Mincha or Rosh Chodesh. This seems to be pointing to Torah reading occaisions other than Shabbat morning.
So why not Shabbat morning? I suspect the answer is a Chabad custom. What is the source of this custom?