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.
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.
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.