I say this as an immigrant: But if you're not an immigrant, a medicare recipient or maybe in the military, your state and municipal governments have significantly more influence on your life than your federal government and most people rarely pay attention to this level of politics.
This lets the people who do pay attention have complete capture. You know your rent is high? Yea that's mostly your state and municipal government doing the bidding of landlords and landed gentry.
High housing prices are probably 5% caused by economically inefficient uses of land (e.g. surface lots in downtown areas) that an LVT would solve, and 95% by zoning and other building regulations. Though perhaps you're right that if the only tax allowed to be collected was LVT, zoning would automatically cease to be an issue.
This lets the people who do pay attention have complete capture. You know your rent is high? Yea that's mostly your state and municipal government doing the bidding of landlords and landed gentry.