The founders,IMHO,wanted a potential potus to at least be born here with parents who were borne here as well to prevent exactly what is happening now.
We have a potus with "Other" allegiances. Zero is so vain and obviously so conflicted that he goes around the world apologizing for what is really his adoptive country and makes us look weak and him look foreign as well as ashamed of America.
This is only one of the many reasons the founders wanted only true American born citizens to be able to run so there would never be an allegiance question.
I think they wanted the president to be born of citizens, even if those citizens were naturalized. That way, there would be no divided allegiance within the family. In other words, I don't think they intended to be as restrictive as to parentage, as you posit.
Before the Wong Kin Ark case, that's exactly the way birth citizenship operated. Those born in the US, but of parents who weren't, for example, amenable to be drafted, did not become citizens themselves. While their parents had to follow US law while here, they had to answer to a call from their mother country, and as such were "under the jurisdiction" of the mother country, but in a different way.