It would be a right if they owned a horse, a privilege if they were forced to rent one from someone else. No one has a “right” to fly commercial airlines unless they own the airline in question. Strange that so many people on free republic apparently don't understand property rights.
The Constitution is silent on the issue (never mind what machinations the SC has to go through to come up with something), so the government has no authority to allow or disallow it. The Constitution does not grant rights (or privileges) to the people. It grants authority in limited areas for the government to intercede in the affairs of citizens.
You said, “No one has a right to fly commercial airlines unless they own the airline in question.”
Do you also think that no one has a “right” to keep and bear arms unless they own a gun factory?