My kids and wife cut my hair every other month or so and I have yet to spontaneously combust. Not sure that fiddling with someone's hair is serious or risky enough that gov't needs to step in the middle.
I'm not certain government intervention is necessary either, but I see ~3 levels of safety. This could be done with a commercial, not government, certification.
Level 1: General hair musser. Knowledge necessary: Be able to recognize the signs of lice, nits, and other hair associated nasties they don't want to mess with. Know to wash their hands and dump their combs, brushes, and other tools into the blue sanitizer between customers to avoid transferring said nasties.
Level 2: Hair cutter - they've been trained on the ins and outs of avoiding cutting the skin while efficiently mowing the noggin in the agreed upon pattern. I don't see this as being hard, so commercial certification is more to show that they can give an acceptable ~$10 haircut.
Level 3: Full service - certified for the safe use of the chemicals used in bleaches, dyes, perms, the use of heat beyond a blow dryer, etc...
As I understand it, hair dressers are to dyes and such as licensed pyrotechnics are to fireworks. They have access to ones that can and WILL cause serious damage to a person if used incorrectly. As such, I have no problems with those working in a Salon to have extra training and certification. But the amount needed in a barber shop that only cuts hair should be less, and the amount needed to not even cut hair reduced even from that.