Aliyah Letorah
The subject of going up to the Torah as a pivotal part of the Bar-Mitzva ceremony is in reality separate from the discussions whether a child can go up to the Torah as any other member of the community. This is because going up to the Torah is essential to Jewish life, it is the spring from which the Jews draw their strength. This might be the reason to say Shepetarani when a Bar-Mitzvah boy goes up to the Torah, since at that time the parents see that he is making good of himself and need not worry that he will be caught up with sin.
Even in communities where children younger than Bar-Mitzvah do go up to the Torah, at the time of the Bar-Mitzvah ceremony much fanfare is made. There might be two reasons applicable to this: since the child is now becoming an adult member of the community and all are celebrating, going up to the Torah is a high profile position, much to do is made (throwing candy and the like); those who do not permit children to go up to the Torah make a fuss and others who permit have copied the others.
Sephardim celebrate on the day a child first puts on his Tefillin which may be several months before he is thirteen. At this time he also goes up to the Torah with all the fanfare of a wedding ceremony.
Ashkenazim generally do not permit a child to go up to the Torah at all until the day of their Bar-Mitzvah at which point it is his right to go up and this pushes aside almost anybody else who may have needed to go up.
So far do Ashkenazim take this custom of not permitting children to go up that when Rabbi Yozfa Shamash mistakenly prepared the portion of Tetzaveh which fell out before he was actually thirteen, the Rabbis of the town did not allow him read but made him prepare the portion of Kitisa.