“If they make it a condition of getting a business license, they have a regulatory mechanism to force it.”
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That is a good point, but I think it could easily be challenged in court.
Is there any state or local government that can set
minimum wage standards for private businesses?
Just about every state has it's own minimum wage laws that are independent of federal law. It generally means the state rate will be higher than the federally mandated minimum. Unless a state has pre-emption laws in place to prevent a county or city from setting a minimum wage, they would also have the ability to demand a value equal or higher than the state prescribed rate. It is very common for cities to impose local sales taxes on top of state sales taxes.