Absolutely! A private business should also have the right to hire or not hire anybody they want,and to admit or refuse admittance to anybody they want.
On the other hand,the airlines are a PUBLIC business. They provide a essential service to the public. They are even subsidized by the feral gooberment to a certain extent. This all puts them in a quasi-government agency catagory,much like universities. Public businesses do NOT have the Constitutional right to deny the public their rights under the Constitution. For instance,if they got together and decided to deny ticket sales to Cuban Americans or American Indians,you and I would not be able to fly. That is much more serious than not being allowed inside certain bars or restaurants.
An individual can also refuse to allow you to enter their home while carrying a weapon.
You betcha! "A man's home is his castle",and all that.
If the FAA lifted the ban on passengers carrying a weapon on board a flight, but American Airlines said that in spite of the ban being lifted, they would not allow passengers carrying guns to board their planes, you will not get on their planes.
That all a theory,and it's totally irrelevant because the FAA is never going to lift the ban,