That way lies a very ugly argument about who is enough on the team. One that almost nobody who thinks themselves American wins, because the real winners of that argument should be the folks stuck into reservations by the alien ancestors of those who see themselves as "true Americans, born of Americans."
For Americans in particular, the best strategy for not having their own legitimacy challenged is definitely not to pull too hard on the legitimacy thread.
In fact, all immigrants who came into this country, actually passed tests (education, work exp, english, etc) before being allowed in. Even in Ellis island at the height of the immigration boom, immigrants who could not support themselves, were sick, etc, were turned around [1]
So yes, we are a nation of immigrants, not a nation of illegal aliens
Except our nation’s shared history, values, and national culture is that we’re a nation of immigrants, a melting pot of global cultures, a refuge for those in need, and a place where anyone can come to seek their fortune, so obviously American parents haven’t been passing on those values to their children if we’re still having this debate, and I think the only fair response to that is to deport all the children who don’t meet your standards of citizenship, by which I mean the entire cohort that’s arguing all this is OK.