They absolutely are aircraft and governed by the FAA. They call them Unmanned Aerial Vehicles (UAV's). FAA also has jurisdiction over all airspace in the US starting at ground level. There's plenty of reference material on the subject you can read.
Sometimes (often?) technical legal definitions considered without context lead to absurd conclusions that don't manifest in practice.
Does the FAA have jurisdiction over paper planes? What about the airspace in my house and does it matter if the doors are open?
On the other hand, I don't think government agencies have much incentive to preemptively limit the scope of their authority, but would be happy to hear of counter-examples.
