I find John Jay to be persuasive myself [post 262].
If you interpret Jay's letter in light of "common law," you see that he's referring to "natural-born" as in, you had to be born here. Chief Justice Waite himself said that at "common law," there was no doubt that if you were BORN HERE to CITIZEN PARENTS, then you were a "natural-born citizen" but there was some doubt as to whether you needed to have citizen parents.