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.

But the airlines don't really give a crap, southwest started basically as an air bus, show up buy a ticket get on. No reservation, no id, nothing.

The airlines don't even check ID most of the time with these electronic boarding passes if your not checking luggage.