This is just the most astonishing claim that I've heard in association with this issue. Not that you're the only one to have made it; you're not. But, it's just nonsensical.
It's Vattel who laid out the requirements for natural born citizenship in a constitutional republic. Samuel Adams, John Quincy Adams and James Otis are recorded as having cited Vattel's Law Of Nations as early as 1764. Benjamin Franklin held Vattel in very high esteem. Our first Chief Justice John Jay likewise.
The meaning of the Constitutional term of art "natural born citizen" was clearly known and understood by the framers; they had a copy of Vattel with them at the original Constitutional Convention.
Because many of our legal traditions are based in English common law. United States common law has its basis in English common law. As the country was very young at the time of the ratification of the Constitution, most of the concepts would be directly borrowed from the English.
It is in no way nonsensical to claim that we have a common law system, we do. It is in no way nonsensical to claim that it is based on the English common law system, it is.
I can’t imagine the Framers were setting out to reinvent the wheel in regards to our common law. It is from the English. Like it or not.
You say they have quoted Vattel, OK, did they quote him on this specific subject saying that is where they got the meaning. I would presume there were a number of books at the convention. Madison asked Jefferson to send him many books from Paris.