Our law was founded on the principles of natural law, and as such all good and proper laws will comport with natural philosophy. Our own citizenship laws, as a reflection of natural laws, recognize citizenship at birth through both blood and soil - but has never established a criteria for both.
Vattel wasn't the first and last word on what “natural law” entails. He is not a secular saint with his words elevated to canon law beyond reproach abridgment or amendment.
And he never did use the term “natural born citizen”.
Obviously Vattel didn't write his legal treatise to state "natural born Citizen." I mean, seriously. He wrote the thing in French, which reads:
Les naturels, ou indigenes, sont ceux qui sont nes dans le pays, de parens citoyens
Translated to English, gives this:
the natural, or indigenous, are those born in the country, parents who are citizens