1 Part of the question concerning Vattel is, “When the writers of the Constitution wrote NBC, were they thinking of Vattel or common law?”
NBS was a standard legal phrase in English law, and multiple courts have found that the meaning of NBC was carried over from NBS.
Had they been thinking of Vattel, they would have written, “He must be indigenous...”, that being the phrasing used by Vattel.
2 Suppose, for the sake of argument, that Vattel was commonly translated NBC prior to the Constitution. One would still need to ask if NBC is rooted in common law as a legal term, or if one sentence in a philosophy book is the limit of what was intended. And to date, the courts have been pretty consistent in looking to common law - where it was an established phrase - rather than one sentence in a book.
Vattel never tried to address what it would mean if an alien father had a child by a native woman, and the child was raised as a native independent of the absent father. That wasn’t his purpose in writing.
Philosophically, a native or indigenous person is someone born in the country of native parents. He was trying to show that meant the natural allegiance was to that country. He wasn’t concerned with Indians, illegitimate children, anchor babies, tourists passing thru, etc. He didn’t need to be. He wasn’t trying to write law.
To remove Obama, you don’t need a little evidence, you need enough to overturn an election based on a fact that everyone knew, and no one with standing (legislature, state official, Congress, other candidates, etc) thought disqualifying.
To prevail in court, you need to show that Vattel was not AN influence, but the only possible source of meaning - and I think you will admit that is not the case. No court will touch it if they have an out. And they do - an interpretation prevalent for over 100 years, rooted in the common law that preceded the Constitution.