You’re entirely correct. My only concern is that tying the federal minimum wage to a less important standard might encourage employers to manipulate that standard in order to keep wages down. Something like the Cost of Living Index (CLI) is major, and really hard to mess with, especially without having many complex consequences. In my eyes, that gives it some built in security. Plus, existing minimum wage laws (whether they’re state or federal) don’t take location into account.
Still, though, I’m not very knowledgeable about economics, and you raise an important point.