Nice of you to admit that, now follow the ramifications through ...
I see no ramifications. I don’t think they used Vattel’s like the Bible, and I think the folks voting on the Constitution had differing opinions on what the phrase meant. I can find quotes on both sides.
My PERSONAL OPINION is that Vattel’s definition is to be preferred. It blocks almost no one, and clearly is the safest way to interpret NBC.
HOWEVER, I think the courts have steadily drifted towards a definition that includes anyone born in the USA who isn’t the child of visiting diplomats or military. And I think any court asked the question now will almost certainly decide that Obama’s mother plus his birth in Hawaii (if it happened) is sufficient.