The employee and the employer are owed nothing by the customer, even though prices may be place on a menu of services, each transaction is a voluntary exchange.
A minimum wage is a convention of a fiat currency, which is a convenience of the actual barter process taking place. A minimum wage gives money without the exchange of assets provided in a proper transaction.
Work is performed primarily because we are capable and adequate to complete the task, or personal utility. Whether anyone else appreciates the completed task is their own prerogative. A minimum wage law assumes the only we can relate in society is our common currency.
It isn't substantiated that altering the meaning and nature of work is harmful. Adam Smith in "Wealth of Nations" didn't discuss the value of the services industry because he couldn't have forseen the rise of a services based economy. Likewise, antiquated definitions of labor input likely do not account for the complexities and the new possibilities of modern economies.