According to your very words, by quoting Vattel, you have shown that Baracks citizenship should “follow the condition of their fathers”...
His father was a British subject.
Barack was not then, and can never be, a natural born citizen of the United States.
By your very words.
Vattel does not determine US law. Also, Vattel was commenting on common practices of various nations, and noted that England - which the US was part of in 1758 - had different practices.
And Vattel never wrote ‘natural born citizen’, which would have been “sujets naturel” in the French, or ‘natural born subject’ in an English translation. Nor did it appear in any translation of Vattel until 10 years after the Constitution - so it is simply false to say Vattel provided a definition of NBC. He never, ever wrote on the subject!