The whole point of a minimum wage increase is that it will increase all restaurants' wage costs to the same degree, at the same time. Meaning that going "down the street", potential diner's will find that all restaurants' prices have increased equally. So the business threat isn't that American consumers will abandon "Dale's" in particular for some other restaurant. It's that they will choose to eat at home instead of eating at a restaurant, or that they will order less food when they do go to one.
"it will increase all restaurants' wage costs to the same degree"
In an ideal world full of law abiding citizens, yes.
In practice, restaurants that cheat are going to win.
I live in a country that has a lot of regulations in the hospitality sector and yet the "paid under the table, not officially employed here" phenomenon is everywhere.