So, to say that "we are a nation of immigrants" is incorrect. That phrase is used to excuse all manner of abuse in allowing outsiders access to and all the benefits of residing in this nation and becoming citizens. Extending citizenship should be selective. We should be encouraging and permitting people to become citizens who are a net gain for the nation rather than a detriment.
The only exception I would grant, would be the genuinely persecuted, and refugees. Refugees would be provided a safe haven until their safe return could be accomplished.
You can immigrate somewhere even if there is no government. I could emigrate to Somalia if I chose to. The nation was founded by immigrants and their descendants. Even the ancestors of American Indians immigrated here at some point. You are splitting hairs that are ultimately meaningless.