Yes, that is one variation. I think the hypothetical is incomplete, and it would be interesting to research how the founders viewed adoption; children born to a woman, fathered by someone other than the husband; and children raised in a house where the father died and the mother remarried.
The general question, IMO, is answered by finding what the founders intended by inserting the qualifier "natural born" only for eligibility to be president. The 14th amendment addresses what it takes to be considered a citizen, but it doesn't mention "natural born" at all.
1. All persons born or naturalized in the United States, and subject to the jurisdiction thereof, are citizens of the United States and of the State wherein they reside.
It does not say:
1. All persons born in the United States, and subject to the jurisdiction thereof, are natural-born citizens of the United States and of the State wherein they reside; and all persons naturalized in the United States, and subject to the jurisdiction thereof, are citizens of the United States and of the State wherein they reside.
I think its inescapable that the phrase "natural born" is intended to probe the allegiance passed to the child in upbringing, and is a requirement peculiar to the offices of president and vice-president.
If you probe the allegiance..then what do you do about :
Bob who was born here of two US citzens and has an allegiance to Saudia Arabia and will bow before the King.
The founding fathers might have thought no such person would be elected.
Isn’t a natural born citizen defined at birth and not what he thinks at the age of 16 and then 20 and then 25, etc.
What’s the definition of allegiance. Allegiance in the mind or merely the country of the father. What if the country of the father cannot be conferred on the child because he was a bastard (as in the case of Obama)