Property tax isn't just about the income to the state. Property tax is about making sure that people are putting the property to work. If you're second mansion is costing you an extra chunk of change every year and you never visit it you'll be more likely to sell.
Additionally, property taxes are (usually) progressive. In places like NYC where they aren't that should be fixed.
Property tax typically doesn't go to the state at all. It's a county assessment. An argument for keeping property taxes is they tend to be more stable than income or sales taxes, and since counties tend to have less float in their budget (a large pile of county spending is on the public school system) it probably works better for them.
The idea of an aggressive mill levy on someone's additional homes seems sane: use it or sell it. I just think universality of property tax isn't actually fair, just like a $200 speeding ticket isn't fair. Indexing it somehow to your worth or income is more fair.
In what way are property taxes progressive? It seems to me that they are inherently a regressive tax (with respect to income on a cohort basis).
(I'm not arguing that they or any other tax ought to be progressive; I'm just not aware of any place where they are progressive on an income-adjusted basis [whether directly or indirectly])
Additionally, property taxes are (usually) progressive. In places like NYC where they aren't that should be fixed.