There has always been some concept of a process by which people can become American and join in that culture through assimilation and integration.

I've had ancestors in North America since 1650, including a vice president and a union admiral, but I also have friends whose parents arrived from Somalia or Vietnam in the eighties and nineties who grew up in largely the same cultural soup I did, speak with the same accent, have the same humor, drink the same beer and eat the same food. Some have served in the US military. In my eyes, they're just as American as I am.