So you grasp the fact that there is no "natural born" statute?
Do you know if there was such a statute then? Believe me, I would be open to a convincing argument that the Founders must have meant something more specific than what they said in the Constitution that they meant.
I really rather suspect that none of them thought it important enough to be more specific. They were dealing with some very important issues in that document. There were some big compromises on the table and while I think that their provision regarding natural born citizen indicated that they wanted there to be some significant connection between a president and this country, I am not sure how much time they spent on the details of that connection. I forgive them if they were distracted by other issues like what to do about slavery, how to protect small states from being dominated by big states, how to balance the powers between the national legislature, the courts and the executive branch, etc.
Our job is to do the best that we can with what they gave us. And, there may not be unanimous agreement about what we should do all the time. I have learned to live in an imperfect world where there are some doubts and uncertainties. We have never elected a President who was without significant roots and attachments to this country. It is not like we have been electing foreign princes as our Chief Executive. Ted Cruz may have been born in Canada, but his biographical connections to America are pretty solid.