We have tried this a few times in North Dakota...
Getting closer, but we are not there.
Who comes out against it?? The "education" industry, top heavy with well paid administrators, but backed by national teachers' unions whipped up an astroturf fear campaign predicated on loss of school funding (the lions share of local property taxes).
In reality, the State is supposed to fund the school system here, not property owners.
That has been decreasingly effective at getting two successive ballot measures to eliminate property taxes defeated, but the bottom line is that the measures failed.
In the Legislature, rather than eliminate the taxes completely, the Primary Residence Tax Credit covers up to $1600.00 of property taxes, and the Homesteader's Credit (means tested, based on income), will reduce property taxes by deducting a percentage of the property's assessed value from the equation (100% for those making under $40K, 50% up to $70K) if you are over 65 or permanently disabled regardless of age (only one of the persons resident can apply). There is also a program for veterans with a service connected disability of 50% or more or an extra schedular rating which results in the veteran being paid at the 100% rate or a surviving spouse, which provides some relief.
The concessions which have been made in North Dakota lessen the impact of property taxes (this past year, I paid special assessments, but no actual property taxes).
(Unlike some states, there is no personal property tax here, only the taxes levied on a fraction of the assessed value of real estate.)
In so doing, the perceived need to eliminate those taxes completely has been muted, and I am ever cognizant of the fact that if those programs are eliminated, the taxes will be back, at whatever mill levy assessed against property values which have increased to seven times the purchase price of my property.
I worked 47 years here in ND so far, mostly in the oil industry, and am about to start a new job in the patch, and I just hate the idea that some day the house and land I worked hard for could be taxed out from under me or my heirs.