Wrong, Hamilton originally suggested
“citizen at birth” and that was rejected for the more restrictive “natural born citizen”.
Natural born citizens are naturally citizens because they can be no other.
Children of foreigners are born with other citizenships.
That is precisely who the founders were excluding.
http://www.art2superpac.com/issues.html
I don't think that's accurate.
According to Wiki Hamilton's first draft didn't have any eligibility requirements. It was only after the convention that he presented Madison with a draft that he claimed he wished he had presented. This draft contained the born as a citizen language but by then the convention was over.
Jay's NBC language was not a response to Hamilton's proposal and I find it interesting that your source tries to muddy the water and make it sound like it was.