Those armed citizens are not allowed in airports by law. There's a reason airports, along with schools, post offices, the state of California, etc, are called 'victim disarmament zones' by the concealed-carry crowd.
I've heard that argument. For me, it breaks down in that it assumes there's a correlation between the number of people carrying guns, and the number of responsible people carrying guns who will actually react properly when something happens.
I've yet to hear of a lunatic trying to shoot up a police station, a gunshop, or a firing range - places where guns are prevalent. Also, Israel arms their teachers[1] and school shootings have 'magically' gone way down. I'd like to think that the deterrent effect would play a significant role in keeping our kids and ourselves safe, aka the "an armed society is a polite society" argument.
Your examples are irrelevant because they're controlled environments - in a gun shop most of the guns are locked up behind bulletproof glass. The police make certain they're the only ones with guns inside a police station. And nobody would be stupid enough to start something in a place where people are practicing shooting targets with already loaded weapons.
The point about the teachers is a good one - though culturally and politically I don't know if that would work very well in the United States. But to be fair to your point, we would have to also allow the students to carry guns to protect themselves against the teachers, who now represent an oppressive and violent arm of the state after all, and we're down to the same problem - how does one tell the difference between a trustworthy, armed person and an untrustworthy armed person?