The short answer is that the data doesn't support generalizing the motives of American immigrants over 300 years from very diverse social, political, and economic conditions. There is no suitable generalization. The same goes for attempting to describe America's current cultural values as a single group.

Longer answer requires the gradiated initial cultural values held by every significant immigrant group (puritans, slaves, Irish, Chinese... significant defined by impact, population, whathaveyou), determining how resilient those values were when thrown into America's melting pots (assimilated? insular?), and to what extent they influenced the groups around them over time. That's a career question, though - not a HN comment that I'm underequipped to answer.