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.