Given: The minimum wage can be too high or it can be too low. Either of these is a problem.
Given: There is some value, or range of values, in between where it’s not a problem.
Given: Where this optimum value or range of values is might depend on economic conditions.
Given: It is possible for legislators to somehow determine the (or a) good value for minimum wage, but it takes time.
If we accept all of these givens, then it follows logically that the minimum wage should be indexed for inflation. If you don’t, then inflation is guaranteed to eventually get you to a point where it’s too low. You’re not going to get legislators to pass a new bill every year to increase the minimum wage by 3% or whatever inflation is.
Now, it’s possible that the legislature got it wrong the first time, and set the rate too high. It’s also possible that they set it right at first, but that changing economic conditions (an increase in automation, say, or emergence of overseas economies) have resulted in it becoming too high. Or, of course, it’s possible by the same token that it turns out to be too low. But both of those circumstances are less common than the routine inflation that happens continually. It’s in those special circumstances that the legislature should change the value manually, and just let it go through the routine changes on autopilot.
EDIT:
You regard it as a problem that the Baby Boomers are all retiring now? Should they not be allowed to?