Does James Truslow Adams undercut his argument about the American Dream when he qualifies the opportunity it offers by adding "according to ability or achievement"? Or is he just being realistic?

I believe he is just being realistic. He is arguing that the American Dream can indeed offer opportunities that are not available to people anywhere else. However, this does not mean that everyone has the chance to become equally successful and wealthy. Your chances and your success are going to be determined by factors such as ability, or achievement. It is a more realistic view of what the "dream" actually looks like in practice.