Yes it does. The owner of a property also has subsurface (mineral) rights and air rights unless they have been sold off or "severed".
An exception on air rights has been made for high flying aircraft.
Please learn to read. I said "upward into the heavens". The heavens, while not precisely defined as to altitude, are higher than the altitude of high-flying aircraft. I have no property rights governing air traffic lanes over my home.
What you call "air rights" are what I already described as the right not to have my house buzzed by commercial planes. That is, your only rights above the earth's surface are those necessary to the enjoyment of your possession on the earth's surface. That's been the US law since U.S. v. Causby in 1946. Your rights prevent airplanes from scaring your chickens, making excessive noise, dropping bolts on your head, etc., and that's it.
Hopefully it's clear, though, that this is in fact the correct moral interpretation of property rights. What's incongruous is that you still own the rights to the land under your house, down to the core--except that your grandaddy probably sold those rights to an oil company.