Look at it in terms of people. Somebody who has good ideas and right principles will impress others by acting upon them and living up to them. Somebody who keeps talking about how different and special and unique he is can be really irritating.
American exceptionalism is like the "Anglosphere" -- an idea based on the exceptionalism of the English-speaking peoples. You can believe in the idea and act on it -- it could very well be true -- but if you talk about it too much it will really get on people's nerves.
Moreover, Smith is arguing in a circle. Look at his conclusion: "By returning to the ideas that underlie American exceptionalism, we can again lead the world." But if we are truly exceptional, truly different from other countries, then it's going to be harder for us to "lead" them anywhere.
If we should be leading them anywhere, we'll only be able to do so by talking about what we have in common, not our own uniqueness. If we are truly unique in a good way it will be our example, not our talk of our own exceptionalism that impresses people.
You're talking about the verisimilitude of the salesman, not leadership. About "building dialogue" and all that liberal claptrap. The author is talking about showing people who don't know, how to design a community sewer system and to get their minds around ideas hitherto unguessed-at in their culture, like rise and run -- engineering concepts foreign to them. And concepts unique in the "Anglosphere", like equity, fair play, and equal justice under law and why they are good for you, no matter what dialect your great-grandfather spoke.