They do. In my state you can do a 2 year apprenticeship in cosmetology, but they can be hard to come by. You need to find a stylist willing to not only train you in practical aspects but follow your bookwork and write/grade tests. It's a two-year unpaid commitment. My wife owns a salon and has taken on a couple apprentices. It is a considerable amount of work. Alternatively a beauty school program can be done in a year, then after a six month internship you can get your license. The apprenticeship is the better option but being able to complete your training in a shorter amount of time with a lower barrier to entry is probably appealing.

In Japan you need to do a 2 year course before starting as a hair stylist. In the UK anyone can set up (although they may have difficulty getting insurance so will probably be operating without).

Regulatory capture. In many states the industry successfully lobbied for licensing requirements on the (questionable) basis of it being necessary for public health.

While the basis for regulation makes sense on paper anyone who's done anything with a low margin service industry knows that whether good practices are followed in a particular workplace has almost nothing to do with education (everyone knows how to keep things clean) and everything to do with management (does management actually want these things done or does it consider them a waste of time).