Considering that there have been plenty of more successful teams both in football (NE Patriots - 6 SB wins) and in American sports in general (GS Warriors, LA Lakers, NY Yankees, etc.) in the past two decades.
There's actually a full Wikipedia page detailing how the Cowboys came to be known as America's Team.
The nickname originated with the team's 1978 highlight film, where the narrator (John Facenda) opens with the following introduction:
They appear on television so often that their faces are as familiar to the public as presidents and movie stars. They are the Dallas Cowboys, "America's Team".
Since that original usage, the name has been used frequently on national broadcasts and the name stuck. There has been debate over whether the nickname is deserved as the team doesn't necessarily represent the best team or the most popular team.
It's just the nickname that was picked by one of the team workers for a campaign. Just like how the Mets go by Amazins and Miracle Mets and how the Yankees get Bombers and Evil Empire.
Near the end of the 70's someone said "This team is on TV more than anyone else" and the name was chosen.