The way I've always heard it is that NFL Films noticed wherever they went, there were a large number of people wearing Cowboys attire. No matter which stadium they were at, Cowboys clothes-wearing people were there. It got NFL Films wondering about that. They looked at NFL Properties and found that the Dallas Cowboys, at the time, was number one in merchandising. Jerseys, hats, trinkets, whatever. Dallas was it. So, when the title of the Cowboys season video was due to be named, they came up with "America's Team". Dallas loved it and everyone else got to hate it.
Contrary to popular belief, Tex Schramm didn't start it. But he loved the name and made a mint off it.