Acutally, the right to travel, which included driving a vehicle in the early days of the automobile, was indeed a constitutional right. Such constitutional rights are infringed when we ALLOW governments to license the activity.
As far as flying, if the government were not involved, then flying would be purely a contractual arrangement between you and a private carrier providing a service; not a right. However, you would have the constitutional right to fly yourself if you had a plane; except that flying a plane is now a licensed activity. Arguing the necessity to license such activities as driving or flying is another matter.
Since the government subsidizes virtually all commercial air traffic in one way or another it is no longer a purely commercial agreement between you and the airlines. The Fedgov is a defacto silent partner that has inserted itself in the private agreement between you and the airlines.
As such, the Fedgov gets to have a say in who gets to fly. However, the constitution was written specifically to prevent the Fedgov from violating individual rights and they certainly do have the right to molest you before they allow you fly.
You make a valid point. One of which I had not thought.
However, in terms of your logic, one could say that because our children are sent to govt schools, teachers would have the right to molest them. We do not have a choice but to send our children to school. We do have a choice to use the government school, private, or homeschool.
In relating to air travel, as you say, there is private air travel, which is then licensed by the government.