>Couldn't a private trusted 3rd-party company provide certification to verify my barber went to school for x or knows what he is doing?

This is basically solved by franchising, but that also adds cost to the producer.

But whatever model you chose it puts the responsibility of informing yourself about the safety of your options on the consumer.

People like when their choices are "safe by default" so they like regulation, even if it doesn't accomplish what it states as long as it appears to be effective it's going to be supported. I can't really decide if that's bad or not, in most cases the safety will be there anyway and if it inspires the consumer confidence it allows the economy to function more efficiently.

Government isn't charging outrageous fees. The outrageously expensive tuition is usually provided by a private party, not the state.

Actually, it is often not the government that charges outrageous fees. Very often, a system is in place where you need to get a license from a private provider, and the private providers of licenses charge outrageous fees.

Incidentally, this explains how many of these certification rules are established in the first place: the certifiers lobby for it, because it brings them profits. (I still believe that the majority of certification rules happen because somebody messed up, which led to calls for stricter regulation.)

The government is the trusted 3rd party.

They price gouge because it turns out that they have a monopoly. The private company would also price gouge because they would also be a monopoly - only you would not be able to lobby the private company and vote for a different one...