No, they only needed the constitutional exclusion for those living at the time of the creation of the Constitution to hold the presidential office. It was the only way in which an NBC is different from the larger class of citizen, and the exclusion covered it. Therefore, the 1790 Act was for others, possibly for the children of the founding generation born in unusual times, the transition of a revolution.
The point is, unlike the FR inventors of law, the founders had no problem putting such a thing in statutory form, and calling it “naturalization.”
Peace,
SR
It’s not clear what this post means.