You have the right to try and fly without an ID. The airlines also have the right to tell you to buzz off and get lost and the airport operator has the right to decide they don’t want you in the building and trespass you if you don’t scram.
Public carriers like airlines are not allowed to refuse service for the reason of refusing to show ID.
They can refuse for other reasons, but the are not “in the loop” when passengers currently get screened by the TSA, which is where RealID is “required”.
You have an absolute "right to travel" (see the 14th amendment and other cases as recently as 1999), but you're also absolutely correct that "common carriers" can can refuse commercial service and you can be criminally trespassed from an airport, BUT TSA can not charge you a fee to attempt to fly.
Unlike other service providers, a common carrier by definition cannot refuse service to anyone willing to pay the fare in the tariff. Common carrier laws are some of the oldest consumer protection laws, enacted to protect travelers and shippers of goods against predatory and discriminatory pricing. Federal law recognizes the "public right of transit" by air, and requires boith airlines and Federal agencies to respect it.
This is only true for the "ticket purchase" not for the service itself.
Outside of ERs in exigent circumstances, any commercial enterprise in the U.S. retains the right to "refuse service" though the nuances of enumerated reasons backed by jurisprudence differ by industry and locale.
An airline can not refuse the "service purchase" unless the customer has been "banned" (technically a trespass statute), but they can refuse to "execute service" for a whole host of reasons including unforeseeable "Acts of God", logistics, or simply if the customer is intoxicated.
This is not true, in at least 2 respects: (1) a common carrier has a legal duty not just to sell a ticket but to provide transportation according to the tariff, and (2) this means an airline can't order a passenger not to board, or order them off the plane, unless they have violated some terms of the tariff. Some airlines have tried to create their own no-fly lists, but without opening up another area of discussion these have no more basis in law than the government's no-fly list, and have never (so far as I know) been reviewed by courts.
Intoxication, ill health, threats to other passengers/crew, assault, and battery are just a few of the many reasons an airline is well within its rights to "refuse service" and this includes not allowing the passenger to ever board the plane.
You're correct that any sort of federal no-fly list is not lawful and, so far, there isn't enough court precedence to make a general policy and the ACLU has won some (limited) court victories. IMO, it is unconstitutional.
In practice, a "banned" passenger is trespassed off of airline property (the plane or gate) and that prior trespass is noted in their customer registry. If the "banned" passenger attempts to fly with the airline again, they are advised verbally or with text that they will not be able to board the plane, but they can still book a ticket. The passenger is then "trespassed" (again) at the gate and not allowed to board.
If you are flying domestically, the airline doesn’t care. They know that someone bought a ticket to get pass security and that ticket matched the ID of the person who got through security. They don’t lose money and thier is no increased safety risk.