You may be quite wrong about that. Back when I was researching this stuff, I learned that John Adams was a member of a debating group in Boston. I forget the name of the group, but I can probably find it again. In any case, they loved the first edition of Vattel they received when it first came available in America, and it quite likely fired their imagination to create the United States. There is a section in it that says a Confederacy should be formed out of a collection of independent states. Nobody else in that era would dare to write such a thing!
I don't have time at the moment, but look up the young lawyer's social club that John Adams belonged to back in 1773 or so.
Adams was exposed to Vattel very early in his career.
The Founders knew ancient history. They knew about citizens in Athens and Rome. They also knew English law through Blackstone, as all lawyers in the colonies had. If they had wanted “natural born” to mean something different from what it had meant for generations of colonists, they would have used more precise language to spell it out.
Australia does have a requirement that no one can sit in parliament who is a dual citizen. This causes trouble because so many Australians are eligible for British citizenship through their parents or grandparents. The current prime minister’s father was Italian from Italy. He got around the requirement by claiming that his parents weren’t married and there is no father listed on his birth certificate. No word on whether the birth certificate is fake.