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.
Not really Minor v. Happersett , 88 U.S. 162 (1875) used the Vattel definition:
At common-law, with the nomenclature of which the framers of the Constitution were familiar, it was never doubted that all children born in a country of parents who were its citizens became themselves, upon their birth, citizens also. These were natives, or natural-born citizens
Some authorities go further and include as citizens children born within the jurisdiction without reference to the citizenship of their parents.
The Venus, 12 U.S. 8 Cranch 253 253 (1814), uses the Vattel definition in it's earlier translation, which left indigenes untranslated and may have reversed the terms, depending on which word from the French was translated as "natives" and which as "natural born citizen" in the later translation. Vattel is explicity cited several times in this opinion, Blackstone not at all.
Both those cases predate WKA, and Minor was decided *after* the 14th amendment was passed. In fact the 14th amendment was the baisis of Ms. Minor's complaint. (She was born in the US, was a person and said that Missouri was denying her right to vote. The court ruled that the right to vote is not a privelege and immunity of US citizenship.)
Now those are all dicta, but so is the "English Common Law" definition as applied to the Natural Born Citizen definition.