The fundamental problem is that home owners and landlords both pay the tax. If everyone rents then tenants will obviously vote for higher property taxes to squeeze landlords. If everyone owns their house they obviously want to pay zero taxes on the land. When you have a mix of both then home owners will vote for the interests of landlords who get to squeeze money out of tenants without paying taxes.
I've seen this pattern literally applied to almost every tax situation. Huge oil corporations should stop polluting? It's obviously a case of CO2 taxes but then the small man doesn't want to pay his tiny portion of the tax. They will always pick out the exception on either end. I.e. a poor polluter paying excessive CO2 taxes because he uses some absurdly inefficient way of heating vs a model rich person that managed to get to 0 emissions because he put hundreds of thousands into reducing his CO2 footprint. The vast majority of people do not fit either stereotype.
But I would pair it with a potential solution that hasn't been truly tried: universal basic income paid for by taxing away the profits of rentiers, a full land value tax, that makes the exchange value of land zero:
http://progressandpoverty.org/
This is unfortunate a very politically fragile solution. The first thing that property rentiers do is eliminate their taxes, such as with Prop 13 in California. But, with enough economic education, maybe it will work...
The closest enactment of this that I know of is in a conservative government, Singapore. Nearly all land is public, and only leased. Only structures can be owned on public land. It's not perfect, and the waiting lists for housing for citizens are still too long, and non-citizens are thrown into a private market, but the outcomes are still amazing compared to a similar place such as Hong Kong.