An Airplane is private property. The owner of that property can deny you access to said property for what ever reason they want (including you carrying a gun).
However if you don't let me carry my gun (I don't personally own one but i believe in our rights regardless) YOU have a responsibility for my safety. Airlines do a reasonable job with security. If I walk into a courthouse, I understand I have to check my firearm at the door but again I can trust that I will be secure there. However haven't you noticed that virtually all these mass shootings have happened in "gun free zones"? I think the owner of a property or business takes on a HUGE responsibility when they put up a guns free zone sign.
While I also have right to free speech I realize that that right might be abridged while I am patronizing a business or I am on private property. I can't just walk in to a store and start spouting off the craziest nonsense. If I don't like I don't have to go to that store. However when I am in public I am free to say what ever I want.
You might think I am crazy but if we where allowed to carry firearms on airplanes I don't think we would ever see another hijacking. Yes occasionally individuals might be gunned down but you wouldn't see mass shootings or hijackings.
Guns are not the problem. The problem is our rowdy violent American culture. Even if you took all of our guns away people would still find ways to be violent. As a matter of fact if you goal is to kill lots of people quickly, guns aren't particularly effective. Automobiles or even a gas tank and matches are much better ways of killing lots of people. As a matter of fact our worst cases of mass murder in this country where committed with incendiary devices and bombs.