"[I]n a new research paper by Doruk Cengiz, Arindrajit Dube, Attila Lindner and Ben Zipperer that looks at data during the period from 1979 to 2016 in 138 U.S. states and regions where minimum pay was increased[, t]he conclusion is that low-wage workers had a pay gain of 7 percent after a minimum-wage law was enacted, but there was little or no change in employment."
https://www.bloomberg.com/opinion/articles/2019-01-24/u-s-ec...
Businesses can adapt, like they do when unions negotiate better pay.
If you make vastly more than minimum and are a consumer of fast food, then don't you think that you will get better service from the one person making $15/hr? Plus, they're less likely to spit on the burger.
The thing about the minimum wage is that there is a market determined wage for a minimally effective employee, but if the official minimum is lower, and it generally is these days, business owners and managers will keep trying to hire at the legal minimum instead. You'd think the invisible hand, etc. but it doesn't work that way. The rule is "the law says X, so we do Y, the minimum required".