I thought the primary argument against it was that the alternative is unemployment. You're implying the options are to pay the same person high or low wages. I don't think this is universally accepted at all.
Which is utterly insane when you think about it. The current minimum wage is practically unlivable, and it's supposed to be a good thing for people to make even less just to box-check "employed"?
At least if you're unemployed, it's theoretically possible to put your energies into learning something to get a better job, but impossible to do that if you're burning all your energy for little more than pocket money.
The only people who'd gain from a lower minimum wage are employers, who'd have the opportunity to exploit cheaper labor.
Because a job is obviously more than a "box check". It's an income. Otherwise, anyone could turn their nose up at those soul-sucking low-wage jobs and choose that unemployment alternative willingly.