Germans, Dutch, Norwegians, and Swedes settled the midwest, which was a key to the development of American society and government. They were a backbone of the Republican Party during the Civil War. Iowa, full of recent German immigrants at the time, contributed more soldiers per capita to the union than any other state. To this day, midwestern speech patterns are full of German or Norwegian influences. There's 50 million German Americans--more than half as many as in Germany.
Italians and Irish had major impacts on the culture and politics of the nation's largest cities. They changed the course of the 20th century by supporting the Democratic Party by large margins from FDR until Reagan, which took the country in a very different direction from the original vision created by the British founders. There's more Irish people in America than in Ireland.
Other European immigrant groups had localized impacts, but never shaped the country the way these other groups did.
I was mostly objecting to your simplified division of people into those "correct" groups whose country this is and those who are foreigners that don't belong here (I know you did not use these words, but that's how your comments sounded to me). I come from a place that has a long history of sentiments like that, and as far as I can tell, there was never anything good coming out of them.