Minimum wages also contribute to this: when you remove flexibility from one parameter of a gigantic system in dynamic tension, it can be partially clawed back elsewhere. So if the mix "lower wages but consistent (and often overstaffed) schedules" is prohibited by the wage law, you may instead get the mix "slightly higher wages but hours aggressively trimmed and rearranged".
I understand the "prices will fall" mantra, but cost of living goes up with inflation. One would think that there would be no need for a minimum wage because as cost of living goes up, no one would be paying the 'bare minimum' anymore. The problem is that this is not what we see in practice.
Another consideration is that middle-/upper-class wages would (in my opinion) no fall as a result of minimum wage abolition. This does not have an insignificant impact on prices too (just because cost goes down does not necessarily translate into lower prices, maybe just higher profits).