Except for those who reject all MW, I may be the only one in the thread to advocate against inflation-indexed MW. Let me explain the case.
First note that if all wages and prices are inflation-indexed, then all price ratios would be locked in. If petroleum increased in price by 10% leading to a 1% CPI hike, then all other prices would rise 1%, leading to a 0.9% CPI hike, a further 0.9% hike in all prices, ad infinitum. This would be absurd, unless you imagine that the prices and wages being locked in were in the correct ratios for all future time.
But, you say, you don’t want to index all prices, just MW. But SocSec is already indexed as are, IIRC, many labor contracts and financial instruments, Wages in the tier above MW will also tend to rise when MW rises. One doesn’t need 100% of prices to be controlled before the above pathology arises.
Free markets, in which prices settle to an equilibrium value on supply/demand curves, are best. Sudden jostling, e.g. the one-time addition or removal of certain rents, can be handled naturally and efficiently by market mechanisms. But throttling or straight-lining the labor supply curve with an ill-advised inflation-indexing are not like that sudden jostling — we’d enter the territory of government planning, not free markets.
The argument that MW is too low is based more on workers’ needs than the value of labor; the problem of poverty, and its associated lack of access to childcare and healthcare etc., can be addressed in other ways than a MW hike. By indexing MW to inflation you impede progress toward these alternate better solutions.
Thus, indexing the MW to inflation is a bad idea.
(Despite this theoretical objection I might vote for such indexing in practice! That’s because U.S. politics are now dominated by an entrenched kleptocracy and progressives need to take what meager wins they can get.)