Absolutely not. The burden of property taxes falls largely on the capitalized asset-value of real estate, so a typical middle-class household can offset much of that burden by paying for a smaller mortgage in the first place. And poor renters largely benefit by not having to pay local income or sales taxes, since the burden of the tax will fall on their landlords. Property taxes are in fact quite progressive.
This is not how the rental business works. Property tax is simply a cost passed on to the renter.
In addition, it's erroneous to think that property taxes are shouldered by landlords. They are, in fact, passed on to renters.
For some states it is a flat rate, I know of one.
But to me is the largest issue is funding for public schools and quality. At one time you would leave high-school with an education that was similar to what you get now in a 4 year college.
These days all we are doing is running a kind-of day care until the public school student leaves to work at McDonalds in the afternoon once the School Day ends.
I also remember some Profs saying in many cases, the first year of collage is a re-education of what the student should already know.
There isn't much that is progressive about property taxes. They affect the middle class much more than the poor (who often aren't homeowners) and the rich (who can afford high tax rates).
Texas specifically has some of the highest property taxes in the nation, and that's in part because we have no income tax. Income taxes, though, are actually progressive.
Note that Texas's high property tax rates is one of the significant contributors to gentrification.