Minimum wages are an economically imperfect (as you've pointed out) but politically possible way to put some downward pressure on much, much bigger failures of

Minimum wages are an economically imperfect (as you've pointed out) but politically possible way to put some downward pressure on much, much bigger failures of our species and society to have attitudes and policies around acceptable minimums for basic human needs that are even logically self consistent, to say nothing of enlightened. We can't quite get it together on saying "food, shelter, healthcare are human rights" or it's sinister sibling "we'll let you die in the cold if there's no profit to be had from you". Those are both consistent, actionable policies, but no one wants a consistent policy on this because everyone gerrymanders it dofferently. So we get clunky hacks like minimum wage that are sort of the average of Aspirational Star Trek and Aspirational Blade Runner.