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.
I find it interesting that you want to open up the Oval Office top foreigners, exactly the opposite of the founders intent.