We are not taliking here about native born American ctizens, we are talking about natural born American citizens and we would all do well to avoid conflating these very different terms.
The constitutional requirement for the President of the United States to be a natural-born citizen had one purpose according to St. George Tucker:
That provision in the constitution which requires that the president shall be a native-born citizen (unless he were a citizen of the United States when the constitution was adopted,) is a happy means of security against foreign influence, which, wherever it is capable of being exerted, is to he dreaded more than the plague. The admission of foreigners into our councils, consequently, cannot be too much guarded against; their total exclusion from a station to which foreign nations have been accustomed to, attach ideas of sovereign power, sacredness of character, and hereditary right, is a measure of the most consummate policy and wisdom. …The title of king, prince, emperor, or czar, without the smallest addition to his powers, would have rendered him a member of the fraternity of crowned heads: their common cause has more than once threatened the desolation of Europe. To have added a member to this sacred family in America, would have invited and perpetuated among us all the evils of Pandora’s Box.