The notion that there is any common law principle to naturalize the children born in foreign countries, of native-born American father and mother, father or mother, must be discarded. There is not, and never was, any such common law principle.
This determination has been the basis of US law for about a century now.
We are of course talking about an undefined term that appears in the constitution for political purposes. My thinking is guided by the political situation in 1787. The Convention devised an office that was unique at the time. The holder of that office would not be a sovereign but a chief magistrate who might have been called a governor-general. He would be George Washington and he would not be a foreign prince. Those were two effective ways of avoiding the establishment of a monarchy to succeed that of George III.
That he had refused to act the part of Cromwell and that he had no natural heirs more or less sealed the deal.