To reiterate: the scenario you lay out (taken to the conclusion that eventually all earners experience the same relative change)
is not possible, because increasing minimum wage does not increase monetary supply.
This is the key way it differs from true inflation. Ultimately, those extra wages – both higher minimum wages and, as you point out, higher wages of those with more bargaining power – have to come from somewhere. Ideally, that somewhere is the extremely high earners who have disproportionately gained income and wealth since the 1970s, thus reducing the relative gap in purchasing power between low and high-income earners.
I don't disagree that the effects you describe happen; my point is that, because of controls on monetary supply, they cannot happen equally to everyone. Nor can a minimum wage hike cause inflation on its own. (Of course, one could coordinate a change in monetary policy to effect inflation which negates the change in minimum wage in exactly the way you describe. The assumption is that the government is not so stupid as to enact two self-defeating polices.)
Of course, whether a minimum wage hike is an appropriate way to achieve the goal of reducing income and wealth inequality is debatable, but it does have a (non-transient) economic effect, because the value of money is anchored by other, more influential, aspects of monetary policy than the minimum wage (namely, the federal reserve rate).