The difference is that immigrants to the US melted their culture in with the existing one, creating the America we have today. Everyone adapted. People came to the USA to become...Americans. One can quibble about hyphens but at the end of the day it's always "-American".

That's the model that works.

Other than it's simplistic with a little fantasy mixed in, it works. I doubt if the Puritans came over to become Americans, or the debtors in Georgia. I think the land, available wood, congestion in Europe, religious differences, and opportunity were some of the motivators. Probably the earliest example of 'blending right in' were the Quebec and Montreal French where a chunk of them lived with the Indians.

We have generalisms - "The third generation is Americanized", which doesn't hold true in many cases and we make some cases tourist spots. In the 1940s German was the second most spoken language in the USA.