What is more pertinent is that English Common Law going back 150 years before the Constitution was established stated that a child born in England was a natural born citizen regardless of the nationality of the parents. And the founding fathers would have known that.
Vattel’s principles of constitutional law were entirely different from the British constitution. American colonists attacked the foundation of the King and Parliament’s power, by demanding that Vattel’s principles of constitutional law be the basis for interpreting the British constitution.