However, you are right that it is probably regressive, since the rich normally place more of their wealth percentage-wise in non-cash assets than poor people do.
Second, can you show your basis for thinking that? I would think it's the opposite: the wealthy aren't using the public schools, free health care, etc.
The upper class has their own private version of pretty much everything.
It is the difference between not being able to afford childcare and not being able to afford a second vacation home.
It's fine to think that, but normally when people are talking about tax progressiveness, they are thinking about the fractional tax on dollars, not "utils". Otherwise, you'd also call our existing, progressive taxation scheme "regressive," since it still probably taxes poor people's utils at a higher rate than rich people. And there would probably not anywhere in the world be a tax scheme that can properly be called progressive.
One good rule of thumb, IMO, is that when you want to change the meaning of a classifier such that all extant instances of the classified type fall into a single bucket, that's probably not a useful change. YMMV.
The cash-poor with few or no financial holdings or savings, are largely unaffected, as are those holding non-cash assets (real property, capital, commodities, stocks).
Also I feel that it is immoral to steal from the future generations (my children) via inflation. Cash investments should be allowed to gain value via productivity gains, instead of constantly cashing in our productivity gains for Govt largess. We'd probably have that "4-hour work week" by now if it weren't for inflationary policies stealing gains.
I know benefits are popular, and the gov't recklessness (defense budgets for example) today makes uninformed people feel like it's a gravy train out there...but really it's a house of cards, arguably steered by IYIs (IMHO).
Inflation as a tax is inherently regressive, as the rich are usually the first to hold the newer money, and the poor the last to hold it, long after all the information about the new size of the money supply has already leaked out to the market.
And it is a tax on holding the circulating currency. The rich often trade their currency for ownership in ventures that produce passive income, or, at worst, low-risk government bonds. In order for those investments to sell, they will have to offer a rate of return exceeding the expected rate of inflation. If inflation is used in place of all taxes, then just reading the government's budget is sufficient to predict the rate of inflation.
The poor see annual wage increases near the expected rate of inflation, if they have sufficient labor-market power, or below it if they don't.
Monetary inflation is a tax on holding dollars. The rich don't hold much of their wealth in dollars, percentage-wise. They hold ownership of productive assets and inherently valuable properties. A tax on money doesn't touch those who get all the money they need from ownership of their wealth.
It seems like rapid inflation would really only hurt the poor and middle class.
(inb4: "Do you want Hitler and Nazis? Because that's literally how we got Hitler and Nazis!" /s)