While you're not wrong in practice, it's actually a surprisingly complicated area of law.
The FAA doesn't "own" the airspace, it's a public right-of-way and every citizen has the right to transit it. See 49 USC §40103: "A citizen of the United States has a public right of transit through the navigable airspace".
The FAA gets to set policy on how to ensure safety, just like the Coast Guard sets rules for the safe navigation of public waterways (but neither "owns" the air/water): "the Administrator of the Federal Aviation Administration shall develop plans and policy for the use of the navigable airspace and assign by regulation or order the use of the airspace necessary to ensure the safety of aircraft and the efficient use of airspace."
Now, where it gets complicated is the definition of "navigable airspace". A common definition is either 360 feet or 500 feet above the tallest structure on a parcel of land, but the case law isn't consistent on this - especially when you consider that some aircraft (like helicopters) can legally navigate lower than that. See: https://en.wikipedia.org/wiki/Air_rights#United_States