[it must be targeted to the inflation]
Indexing minimum wage to inflation is a positive feedback loop that is inherently explosive. Once you put a few of those policies in place, say hello to Weimar Republic. Granny git your wheelbarrow!
I use to think as you do but am beginning to understand just how the Fed controls the economy. If they control just how much inflation is to occur, how can you believe that wages should remain the same.
Now in reality, it really hurts the private sector the most because as they increase inflation and the benefits are tagged to inflation, only those in the private sector are hurt. It’s rare that private business keeps up with inflation. They may try, but it is usually minor (I can attest to that in my 40 years of private jobs). They do give appropriate benefits due to capability, experience, and performance though - unlike the union controlled jobs.
While I don’t believe in minimum wages or wage control, I have to take into account the actions of our government. It is not pretty.
If you ignore their actions then not only will private sector be damaged but the social network will also fail.
Just saying that inflation hurts all sectors but hurts those on the low end the most. I’m sure that your also realize that. Yes, that includes everyone on a fixed income and everyone on any government benefit. While I can understand anger at that, those folks are in dire straights when inflation occurs.