I think the attitude you are talking about is largely driven by media.
My country has a $20 minimum wage, and yet the unemployment rates, small business survival rate, and inflation are all around the same levels at the USA, so all the talk of impending economic catastrophe if we give poor people a few more crumbs seems to be hot air from where I'm sitting.
For example: hiring one of two candidates, Nick and Joe, would be profitable under 30$ per hour. Nick asks for 15$, its Joe's first job so he's willing to take 10$, I go to Nick and say, "look Joe will do it for 10, so I really can't justify paying more, but I'd prefer to pay you those 10, since you're experienced". If Nick has a better offer elsewhere, he has no problem. If not, Nick gets just 10.
If the state says 15 is the minimum wage, not only Nick but also Joe must get 15, so my best move is to take Nick at 15. Clearly, a higher minimum wage _can_ benefit the worker, and not only the very weakest one.
Capitalism allows each participant to seek only the best available deal that is agreeable to them, sure, but availability is subject to negotiating power, and that is distributed very unevenly.