I think you two are talking past one another. The Constitutional provision that mentions "a citizen..., at the time of the adoption of this Constitution" is irrelevant these days. It was just meant to clarify that, at the time of the adoption of the Constitution, anyone who was then a citizen of the US qualified as a "natural-born citizen" for purposes of serving as President. At that time, the large majority of American citizens had been British citizens from birth, since the US had not yet been formed. This provision was meant to prevent them from being disqualified from the Presidency.
But this provision is irrelevant to our discussion here.