Even in other states which lack property tax caps, the basic framework within which tax rates are set is controlled at the state level. For example, some economists have proposed replacing property taxes with land value taxes but it would be impossible for a single city to do so without enabling state legislation.

Yeah, here in IL property tax is 100% local (state law sets limits and requirements only). And everything is an independent government body with taxing authority.

The city collects its taxes, the school district collects its taxes, the library collects its taxes, fire district, road department, forest preserve, etc. None of them can tell the other to do anything - they are all independent with independent funding. The county has its own taxes, but also has the function of actually providing the payment, billing and distribution services. So you only have to make a single payment.

By the way, property taxes in Florida work nearly the same as California and they don't have any problem building lots of housing.