Yes, they used Vattel’s book. They did not adopt it as binding law. And the idea that the Constitution or Kent were referring to Vattel’s book remains laughable. What are the offenses against Vattel you would have the Constitution refer to?
What’s laughable and a huge stretch by you trying to turn jus soli births from a bygone era of feudalism, as the same as it is referred to under Article 2, Section 1 of the US Constitution on who constitutes a natural born citizen.