Restaurant employees, preparing or serving food, could spread disease or cause harm, but are not obligated to be licensed. Would you support requiring cooks, servers, wait staff, etc., to undergo licensing?
That said, these licenses are typically about $10 and can be completed by online website in about an hour. I would expect something similar could work for hair cutting as well, but probably not for all the more exotic things like hair dying.
The restaurants require licensing out the ass, and I know there are ABC (alcohol handling) certifications required for servers, but cooks don't. At least one person (typically a manager) needs to be ServSafe certified, but that's not a requirement for every worker.
Meanwhile, a hairdresser must be licensed before they can step into a salon and do work. My sister and wife both worked in the industry, and I promise you they would both scoff at the idea of a 1 hour form/test for someone cutting hair. There's a ton of hygiene related things that they need to know to prevent disease transmission from cutting and handling hair daily.