Many an American with these "Immigrant Roots" you speak of, came through Ellis Island, New York, and went through the legal process in place at that time.
With Official Papers, they were let in, whilst others were Without Official Papers and were detained until they either got some or were sent back to their home country of origin to acquire some.
Some came through Angel Island, in California
When you enter another country, do you need a Passport or work visa for entering and living in their sovereign nation for a limited amount of time specified? Why is that not a reciprocal program to you?
Is it because some Frenchie Artist wrote a poem on a tablet of the Copper Statue of Minerva his nation commissioned and gifted the USA ?
If so, abandon the romance fantasy dream, and figure out the USA is a nation of Laws.