Restaurants and bars are not common carriers, but airlines are. At least under US law, that puts them in a special category. In return for certain benefits (immunity from many lawsuits, enforcement of contract of carriage even when passenger is unaware of them, etc.), they give up certain rights that most businesses have. One of these is the right to refuse service arbitrarily. A common carrier must accept a person's business unless they can specifically demonstrate why they should not.