This is another foundational myth of the USA. Most of the people who moved from Europe to the USA were relatively privileged, particularly early on. Most of the people on the Mayflower were what we would now call middle class. Educated professionals and skilled tradesmen from England. The Puritans who didn’t have money just couldn’t afford to get out of England.
There weren’t large amounts of poor immigrants until the 19th century, and the USA was deeply discriminatory against them when they came.
Also, I think many black folks and indigenous people did not experience “an objectively better life for their descendants.”