That's a nice story, but it's just that. Alternatively, all the minimum wage workers will have more expendable income. Some of that may be spent at those brick-and-mortar shops, and will offset the additional costs.
Remember that the velocity of money makes it a non-zero-sum game.
EDIT: Ends up that Wikipedia covers this nicely:
https://en.wikipedia.org/wiki/Minimum_wage_in_the_United_Sta...
"The interconnection of price levels, central bank policy, wage agreements, and total aggregate demand creates a situation in which conclusions drawn from macroeconomic analysis are highly influenced by the underlying assumptions of the interpreter."
Also has a nice side-bar quote from Henry Ford, who was partially known for higher wages, and which I think addresses the point I was trying to make (emphasis mine):
"The owner, the employees, and the buying public are all one and the same, and unless an industry can so manage itself as to keep wages high and prices low it destroys itself, for otherwise it limits the number of its customers. One's own employees ought to be one's own best customers."