Iceland, Norway, Sweden, Finland, Denmark, Austria, Germany, Switzerland have no minimum wages. They are doing pretty well.
Note that in Sweden unions aren't company specific so there is no way to prevent people from joining one.
Arguing about the legality is a harmful modern fetish that especially in contexts such as these does not help us move along.
Minimum wages set at national or state levels is more common in countries where labor unions don't exist to negotiate.
There is a reason Europe moved past min-wage laws, but Americans seem to be hell bent on not listening. Like most children, they have to make the same mistake on their own in order to learn from the consequences. I have personally given up on trying to reason with people about this.