Kettle, Pot.
I have read the evidence and it supports the idea that someone who is born a citizen is a “Natural Born Citizen”. There isn’t an extra layer to reach that mythical status.
And yes, Vattel was widely known in the time of the writing of the Constitution.
But so was English Common Law.
And English Common Law was the basis of our law, except where specified.
If the writers of the Constitution wanted to use Vattel as the basis for who was eligible to be President, they would have spelled it out, or at least the early courts would have supported it. They didn’t. They supported English Common Law.