Because many of our legal traditions are based in English common law. United States common law has its basis in English common law. As the country was very young at the time of the ratification of the Constitution, most of the concepts would be directly borrowed from the English.
It is in no way nonsensical to claim that we have a common law system, we do. It is in no way nonsensical to claim that it is based on the English common law system, it is.
I can’t imagine the Framers were setting out to reinvent the wheel in regards to our common law. It is from the English. Like it or not.
You say they have quoted Vattel, OK, did they quote him on this specific subject saying that is where they got the meaning. I would presume there were a number of books at the convention. Madison asked Jefferson to send him many books from Paris.
It is very doubtful the framers adopted the doctrine found under the old English doctrine of “natural-born subject.” The British doctrine allowed for double allegiances, something the founders considered improper and dangerous. The American naturalization process required all males to twice renounce all allegiances with other governments and pledge their allegiance to this one before finally becoming a citizen.
Framer Rufus King said allegiance to the United States depended on whether a person is a “member of the body politic.” King says no nation should adopt or naturalize a person of another society without the consent of that person. The reason? Because “he ought not silently to be embarrassed with a double allegiance.” House Report No. 784, dated June 22, 1874, stated, “The United States have not recognized a ‘double allegiance.’ By our law a citizen is bound to be ‘true and faithful’ alone to our government.”
Under the old English common law doctrine of natural-born subject, birth itself was an act of naturalization that required no prior consent or demanded allegiance to the nation in advance. Furthermore, birth was viewed as enjoining a “perpetual allegiance” upon all that could never be severed or altered by any change of time or act of anyone. England’s “perpetual allegiance” due from birth was extremely unpopular in this country; often referred to as absurd barbarism, or simply perpetual nonsense. America went to war with England over the doctrine behind “natural-born subject” in June of 1812.
Because Britain considered all who were born within the dominions of the crown to be its natural-born subjects even after becoming naturalized citizens of the United States, led to British vessels blockading American ports. Under the British blockade, every American ship entering or leaving was boarded by soldiers in search of British born subjects. At least 6,000 American citizens who were found to be British natural-born subjects were pressed into military service on behalf of the British Empire, and thus, the reason we went to war.
Source: federalistblog.us
The answer is yes. There are some - not many, but some early quotations and references - indicating fairly conclusively that the Vattel definition is precisely what the framers had in mind. These can be researched on the web. The Federalist Society is a good place to start.