Not true. They do control what operates in the airspace, in some cases down to ground level. For example, an ultralight aircraft (in the US, below 254 lbs empty weight) can fly just about anywhere, as it's not under FAA jurisdiction unless built or flown beyond FAR 103. If one crashes, the Feds call the local police. 

No one 'owns' their airspace, but local rules and zoning has a big say on building restrictions. And yes, those with the deep pockets usually get their way. 

There's a big housing/condo development here where the builder put in a few big homes on a hillside overlooking Lake Michigan. After those sold to the super-wealthy, the developer then built multi-unit condos in front of the homes for doctors, lawyers, etc. That's where he really made big $, but ticked off the first buyers. They didn't 'own' their views.