Please don’t confuse the celebration with the cutting. If you cut the hair on the third birthday and you make a celebration on that same day, it is not necessarily for the haircut. The Machzor Vitri has a long piece about the day of bringing the child to Torah. He mentions many Minhagim there. So, in essence it is a sub Bar Mitzva celebration.
Also, the Taz says not to teach Torah (or Alef Bais, if I’m not mistaken) to a child under three. He aludes to the Pasuk about Orlah saying, Three years it shall be Orlah.