Citizen at birth was rejected by the founders as insufficient. The chose natural born citizen specifically to exclude the children of foreign nationals.
Alexander Hamilton’s suggested presidential eligibility clause:
No person shall be eligible to the office of President of the United States unless he be now a Citizen of one of the States, or hereafter be born a Citizen of the United States.
Many of the founders and framers expressed fear of foreign influence on the person who would in the future serve as President of the United States since this particular office was singularly and uniquely powerful under the proposed new Constitution. This question of foreign influence was elevated when John Jay considered the additional power granted to the Presidency during times of war, that is when he serves as Commander in Chief of the military. Jay felt strongly that whoever served as President and Commander In Chief during times of war must owe their sole allegiance to and only to the United States.
Because this fear of foreign influence on a future President and Commander in Chief was strongly felt, Jay took it upon himself to draft a letter to General George Washington, the presiding officer of the Constitutional Convention, recommending/hinting that the framers should strengthen the Citizenship requirements for the office of the President.
John Jay was an avid reader and proponent of natural law and particularly Vattel’s codification of natural law and the Law of Nations. In his letter to Washington he said that the Citizenship requirement for the office of the commander of our armies should contain a “strong check” against foreign influence and he recommended to Washington that the command of the military be open only to a “natural born Citizen”. Thus Jay did not agree that simply being a “born Citizen” was sufficient enough protection from foreign influence in the singular most powerful office in the new form of government. Rather, Jay wanted to make sure the President and Commander In Chief owed his allegiance solely to the United States of America. He wanted another adjective added to the eligibility clause, i.e., ‘natural’. And that word ‘natural’ goes to the Citizenship status of one’s parents via natural law.
Below is the relevant change to Hamilton’s proposed language detailed in Jay’s letter written to George Washington dated 25 July 1787:
Permit me to hint, whether it would be wise and seasonable to provide a strong check to the admission of Foreigners into the administration of our national Government; and to declare expressly that the Command in Chief of the American army shall not be given to nor devolve on, any but a natural born Citizen.
http://www.art2superpac.com/issues.html
“Natural law” and Vattel’s codification are IRRELEVANT. The supreme law of the land, the Constitution specifically enumerates that power with Congress.