>Private organizations could do the same safety checks--plus, they would have an incentive to do only the checks that actually added value,
Private organisations have an incentive to increase profit, even if it means a loss of public safety.
Accountability only provides economic leverage when legal remedies for tort and malpractice are cheap and easily available.
If they aren't, private organisations have no incentive at all to act responsibility. Often they don't.
I think hairdressing may be a special case because salons serve a local area, and it's a very social profession, so bad word of mouth can do a lot of damage.
But that's not true of most businesses. In fact of the drivers of the corporation as a business model is exactly the freedom it provides from personal and social responsibility. Instead of being personally accountable to customers, larger corporations operate at a level where local reputation is meaningless, and only governments, class actions, and the occasional PR disaster can punish bad faith operation. (Of all of those, PR disasters are often the least damaging.)
There are licensing requirements for corporations trading in specific fields and for specific products. But there's no such thing as a valuable and reassuring general certificate of corporate professionalism and responsibility - which is perhaps a strange thing considering the power and influence corporations have.