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.
“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.”
Here’s one thing I’ve wondered. If the framers of the 14th amendment truly intended to exclude those with divided loyalties from being citizens, why didn’t they specify “not under the jurisdiction of any other nation,” instead? To the other country, dual citizens would appear to be under the jurisdiction of the U.S., as much as we might say a kid with a German citizen father is under German jurisdiction. One jurisdiction doesn’t cancel out the other.
If the amendment said under the jurisdiction of the U.S. and only the U.S., I’d undertand. But for now, I don’t.