If "natural born citizen" required both parents to be born in the US, this clause would be insufficient, because it would not cover the generation of people born in the US to parents who themselves had been born under colonial rule. For example, Martin van Buren (born 1782), son of Abraham van Buren (born 1737) and Maria Hoes (van Alen) van Buren (born 1747) would be ineligible, as neither Abraham nor Maria were born in the US. I haven't been able to trace the birthdates of both parents of John Tyler (born 1790) or James K. Polk (born 1795), but they, too, were almost certainly born to parents who had not been born in the US.
Obviously, the Founders did not contemplate this definition of "natural born citizen".
It didn't. It just required that the parents be "citizens" at the time of the child's birth -- born, naturalized, or grandfathered in.
I disagree and do not find the line of reasoning you used to be persuasive.