>> Vattel was referring to natural citizenship at birth. It stands to reason why this would be translated as natural-born citizens. <<
Except it wasn’t, historically. It was translated as “natives.” And the point is that Vattel was therefore NOT what the founders could have been referring to when they used the term, “natural-born citizens.”
In fact, Vattel was commenting on civil law in Romanist (that’s a system of law, nothing to do with Catholic) legal cultures which do NOT automatically grant citizenship based on land of birth, but based on father’s land of birth (which is why the entire Italian hockey team a few olympics ago was born in Brooklyn and similar places; U.S.-born children of parents from Roman-law countries are still citizens of those Roman-law countries.
This means that the founding fathers could NOT have been basing their concept of nationality on Vattel.
Sorry, but you're only looking at one word in the overall passage about "Citizens and Natives" as translated from the French words "Des citoyens et naturels" ... It is talking about citizens and natural citizenship at birth through their fathers.