I think it's important to remember also that they don't just do hair. They use makeup, and I'm sure for liability reasons they don't want that makeup on a child under 3's face. It's near their eyes too, so I personally would worry about my child rubbing it into their eyes at that age. Also, that's a TON of hairspray and stress to be putting on a young child's hair. The age of three isn't arbitrary. They pick an age where they feel the child is mature enough to listen to parents, physically grown enough to not have severe skin issues caused by makeup and with hair thick enough to handle the stress they put on it. Not to mention, again, this could be a liability thing depending on what products they use.
IMHO, go to the Harmony Barber Shop. It will be more unique, less strain, no makeup and faster, not to mention a fraction of the price! Also, when the Fairy Godmother in Training asks your little one how old she is, what will she say? If she says two, are you willing to risk being asked for a park ticket? For the record, I've been to the BBB, and this is a question asked (not in a forceful way, just in a getting to know you way) before they start doing anything (as the FGMT combs through to check for lice, etc.).
