>not have to compete to attract employees. Also strongly implied is that minimum wage workers do not have an alternative means of supporting themselves.

In practice, that's the case. I think that the idea that it isn't so, and people go to this or that employer out of "free choice", stems from a religious-like belief in "free market", than from real-life empirical evaluation of how desperate and poor people pick jobs - and what options are open to then.

>Still, these people will still have to contend with employers that treat them like commodities, with no real alternative way to support themselves.

True. But at least they will make some more money will doing so - and might even afford to save a buck or too, and eventually look elsewhere.