Let's leave the issue of royalty, with the possible complications of claims on the British throne, out of it for now.
Suppose a British general who becomes dissatisfied with the regime moves to America, marries an American woman and sires a son.
That son is raised as American from birth and never feels loyalty to any other nation.
I think the founders would relate and I see no reason they would want to prohibit that boy from becoming President one day.
In both cases, the son born could be American or British. A natural born citizen can be no other.
A son born here of American parents can be nothing but American. No other country on the planet can call him a citizen at the time of his birth.
This is what makes him a natural born citizen as opposed to a citizen.
It is also why the “Citizen at the time of the adoption of the constitution” is in the text, not one of the founders was a natural born citizen.