Good question. There aren't many songs that specifically talk about competition, self-reliance and hard work. In music, those things are often more implicit than explicit. "Freedom" is common to talk about, but that's because it's a more general term which can be used in many different ways.
The famous patriotic song "God Bless the USA" by Lee Greenwood has some of those qualities, but the song is more about loving the USA than talking about individualism and competition.
There might be some country songs that are good examples. Right now, the only one I can think of that might fit is "Small Town Southern Man" by Alan Jackson. It tells the story of a simple American farmer who works hard raising a family and is grateful for everything he has. In many ways it typifies "traditional American values," even if it doesn't specifically talk about things like competition and freedom.