I have seen no proof that “natural born citizen” was not used by the founders as an alternate way of saying “native born citizen.”
We have zero documentation of how the wording wound up as it did. This portion of the Constitution seems to have been voted in with almost no discussion.
One is either born, and via that natural act, has citizenship under natural law - and is thus a “natural born citizen” - or one must go through a legal process to confer what nature did not - they mus be “natural-IZED” into citizenship.
There are only two ways of becoming a U.S. citizen - and it makes sense to me that there are correspondingly two types of citizens of these United States - those who are natural born citizens - and those who had to be natural-IZED.