This is a great question and I like how you ask it. No, the American dream is not about simply gaining fame and wealth, though it has come to mean that for many people.
The American dream in its purist form is much more noble. It is about people coming to America for opportunities. All you needed was the heart to work hard and carve out a living for yourself and your family. In other words, the American dream stated that hard work, industry, and faithfulness will pay off. Based on this ideal, often times people equated fame and wealth with success and happiness.
With that said, the American dream had other elements are well, which were important. Many people came for education, freedom of religion, and other liberties that America offered.