City and state governments have no legal jurisdiction over aviation and cannot enforce any and all local ordinances in regards to aircraft. Only the FAA has the legal authority and oversight authority of any and all aircraft, including drones. Make sure you understand the law. City, state and county laws and ordinances are essentially worthless, hoping you (the resident) are none the wiser.
I believe you're mistaken. The FAA indeed is the authority on all aircraft, while they're in the air. Once they land, they enter the jurisdiction of whatever state, city, county and whatever ordinances they wanna come up with. As for private airfields and airstrips, the FAA is only the registrar of record to include them in navigational charts and document any obstructions, if any, and does not "allow / deny" their creation, only marking them as "objectionable" if their operation is risky, as long as they're a certain distance from certain classes of air space (like class Bravo for commercial airports). Cities and counties can (and usually do) have ordinances against private field operations, noise thresholds, having fuel on-site, etc.
