The thing is, you assumed what I believe. I'm actually in favor of higher minimum wages and indexing to costs. But the feedback-loop nature of the economic system, or any feedback system (where future output numbers of the system depend on previous outputs), must be acknowledged and accounted for, or else you will, mathematically, get exponential growth and/or oscillation.

I also have a lot of family who currently live just fine making less than $15/hr. Their employers (small businesses) would have to raise prices (or go out of business, or both), which would raise costs, which would raise the CoL index, which would raise future minimum wages in a feedback cycle, if there is not some compensating factor in how wages and prices are set.