Very few of us can actually track our ancestors back to the original colonies, so I'm not sure that's a relevant point.
An appreciation of American culture, liberty and the Constitution is not a genetic trait. Native-born Americans used to think that Catholics couldn't be loyal Americans because their true allegiance was to the Pope. Anyone can be assimilated into believing in American values, whether they came here legally or illegally. We just haven't been putting enough emphasise on assimilation.
All the other stuff, like preferences in food, ethnic origin or America's favorite sport, change over time and have little, if nothing, to do with American culture. To quote the Sopranos: "Until Italians got here, Americans knew s**t about food."
And, anyway, I like tan women.