This whole issue has nothing to do with whether or not you have the right to demand that a common carrier transport you or not (although, contrary to what you wrote, you generally have that right as well). This has to do with whether or not the federal government has the power to restrict you from getting the service that you and the common carrier agreed to. That power is nowhere in the constitution.
By the way, the fact that the common carrier has already accepted payment for the service, and that the traveler, in his or her capacity as a taxpayer, has already paid for whatever part of the air travel system is funded by government, are just gravy, and serve to point out that the government is not only restricting you from making a deal, but they are preventing unsuitably obsequious persons from getting the benefit of whatever deal the traveler already paid for.
We are in agreement. It is government interfereance with a private contract that is the problem here. In fact, this is just a further errosion of the freedom of association due to government regulation. It has nothing to do with a “right” to travel on a commercial carrier.
It would seem to me that the air carriers should be doing this themselves, perhaps under some kind of Federal oversight if it is deemed necessary as a legitimate function related to interstate commerce.