Takahashi involved a person born outside of the U.S. who later came to the U.S. with the hope of becoming a citizen. So the case is irrelevant to the question of the status of a person born in the U.S.
In 1898 the question of the status of a person born in the U.S. to parents of foreign nationality was addressed. In that case, U.S. v. Wong Kim Ark, 169 U.S. 649 (1898), the court rule that the phrase "born . . . in the U.S., and subject to the jurisdiction thereof" in the 14th Amendment took its meaning from the common law meaning of "natural born citizen," which in turn derived from the English common law term "natural born subject." In that case, the U.S. government and dissenting justices urged application of Vattel. Vattel was rejected as being a source of the applicable rule.