Look at it this way, can, or should, the state be able to override a county's ability to limit a cities property tax?
In the United States? Absolutely. That's how our system is structured. The States retain all powers not specifically surrendered to the Federal Government, per the US Constitution. They are sovereign entities in their own right, not dependent on Washington for their power. Their political subdivisions are completely arbitrary creations, that can be created or destroyed at the whim of the State Legislature.
If push comes to shove, what happens if NC or TN simply dissolve the political subdivisions that are attempting to do this? Will the FCC also try and prevent that? Where would that authority come from?