"John Jay wrote in a letter to George Washington dated 25 Jul 1787:
"Permit me to hint, whether it would be wise and seasonable to provide a strong check to the admission of Foreigners into the administration of our national Government; and to declare expressly that the Commander in Chief of the American army shall not be given to nor devolve on, any but a natural born Citizen. ""
So where does John Jay say "natural born Subject"? No, he does not.
He used the phrase “natural born citizen” because we are a REPUBLIC and not a MONARCHY! It substitutes ONE WORD because that allows a common, well known legal term to transfer over from English law into American law.
What did Jay MEAN by NBC? Well, since he is using a phrase derived from English common law, he means the same person who would be a natural born subject if in England would be a natural born citizen if born in the USA. And that means people having alien parents - if they are here ‘in amity’ - would be natural born citizens in the US, just as they would be natural born subjects if born in the UK.
We do not fall under English common law, but it provided the language used by all colonial lawyers at the time the Constitution was written. That is what Jay was thinking, because he was writing a legal document. English common law provided the lingua franca of colonial lawyers.
It wouldn’t matter if he had memorized every line in Vattel - when writing law, he would use the language of English common law to express his ideas.