Anyone who has any professional familiarity with U.S. legal history knows that the common law of England was also the common law of the colonies, and remained the common law of the states thereafter unless superceded by statute. To argue that the ratifying states would have been endorsing an interpretation of a foreign legal system (the continental system, via Vattel), over the English common law view -- which common law was practiced every day in American courts -- makes no sense.
Our legal tradition is the English common law. If the Constitution is silent in terms of defining a term, resort to English common law interpretations would have been the natural instinct and belief of Americans at the time of ratification.
I’ll add that Vattel’s Switzerland still bases citizenship on who your parents are, not where you are born.
“Swiss citizenship is propagated by Jus sanguinis. A person is a Swiss citizen at birth (whether born in Switzerland or not) if he or she is:
born to a Swiss father or mother, if parents are married
born to a Swiss mother, if parents are not married
Where parents marry after birth and only the father is Swiss, the child acquires Swiss citizenship at that point...
Jus soli does not exist in Switzerland, hence birth in Switzerland in itself does not confer Swiss citizenship on the child.”
http://en.wikipedia.org/wiki/Swiss_nationality_law
http://www.eda.admin.ch/eda/en/home/reps/ocea/vaus/ref_livfor/livaus/swicit.html