I don't give a flying F about what you don't give a flying f**K about! I do care greatly about enforcement of oour Constitution! Every jot and tittle of it!
No, actually, you don't care about enforcing the Constitution, you care about finding some mostly irrelevant hook to "justify" the end-results you want to reach for other reasons. Vattel wasn't elected, didn't write the Constitution, or any part of it, and wasn't expressly incorporated into any part of the Constitution by express or necessary implication.
Vattel is not determinative of what the terms of the Constitution mean, including the requirements for being president. Vattel can be used as guidance to try and work out a principled distinction between the bare term "citizen" and the phrase "natural born citizen" and the simplest, and most straightforward, back in the day when there weren't even any simple immigration controls, was that the Founders were looking for some proxy, some indicia that an individual was more likely not a foreign mole, someone with foreign loyalties who was attempting to become president to subvert the U.S. The indicia they arrived at was that if someone had been born in the U.S., and had grown up as always being an American, that such a person would have the requisite lack of foreign entangling loyalties. Thus, a "natural born citizen" within the meaning of the Constitution most comfortably fits with the distinction between an individual who was born here and an individual who was born elsewhere and subsequently immigrated to the U.S. That this is also consistent with the express language of the 14th Amendment further buttresses the interpretation of the phrase "natural born citizen".
On that basis, the most reasonable reading of the requirement that the president be a "natural born citizen" is that the person have been an American since birth - that he or she have been born in the U.S. - and not merely naturalized into U.S. citizenship when he or she was already an adult (minor children cannot become naturalized citizens before they turn 18).
That is a sufficient distinction between the bare term "citizen" and the compound phrase "natural born citizen", it accomplishes the purposes the Founders appear to have had when they drafted the requirement, and is therefore the most likely interpretation of that phrase in the context of who can be citizen.
Trying to impose Vattel as the end-all-be-all of the meaning of the Constitution is more of an abuse of the Constitution than finding a workable, reasonable interpretation of the language is.