It doesn't go nearly as far is it should, indeed, and at least in Texas, you can't claim the homestead exemption the first year you own a property. So I bought a house from someone who had been claiming the homestead exemption and also a senior discount. I immediately filed for the homestead exemption as well, but that doesn't take effect until the second year you own the house. So the property take history looks like: very low, very low, very high, somewhat low, somewhat low.

That rule about paying the full amount on year one is... odd.