I understand what you are saying. However, I believe your conclusion is absurd. Considering most of the US does not require licensing for a "shampoo technician", it would appear you are arguing for something that is currently a fringe practice to become mainstream. Is that correct?
Also, a cosmetologist and a shampoo technician are two distinct positions.
I'm not arguing for more or less regulation. I'm arguing that we have a societal interest in having some direct or indirect regulatory handle on many jobs that don't obviously require it, and cases like you're raising of the dishwasher, where regulation seems obviously unnecessary, are in fact indirectly regulated--and the costs of regulating the dishwasher, among other restaurant employees, are actually quite significant (meaning that '300 hours' of regulatory training, appropriately targeted at the business owner, isn't as obviously unreasonable as it appears).
You're holding up 300 hours of training for a dishwasher as a reductio ad absurdum of professional regulation of hair washers in salons, when what's obviously happened is a local regulation saying that salon 'workers' (again, actually independent contractors) need a diploma from a cosmetology school. 300 hours is less than 10 weeks of full-time school; a 3 month program teaching salon workers how to effectively, hygienically and with due respect for local business regulations, work in a salon. When you see 'even the hairwasher needs a cosmetology diploma', it's closing a back door where the not-a-cosmetologist hair washer is allowed to do all the things a cosmetologist does for cheaper--and without the required training to actually be an independent contractor.
I'm coming at this from the perspective of someone on my condo building's strata council (i.e. HOA). I'm continually dealing with the push and pull of rules at the most petty level, part of which means people being shockingly petty about avoiding the plain intent of a reasonable rule. You'll never see the truth of the saying 'decisions are made by those who show up' as you will in a minor bureaucratic positions like strata/HOA. It's a constant battle against senior citizens who want your garbage bagged up into one of eight separate classifications and the guy who argues that his truck that hangs six feet out of his stall in the parkade isn't a problem because the rule says "don't block the lane" without specifying by how much.
Also, a cosmetologist and a shampoo technician are two distinct positions.