No, because we are not discussing if English common law applies to the USA. It does not.
However, English common law DID provide the language of law for the Founders. Their understanding of what a legal term meant was shaped by living under and using English common law every day of their lives.
They could have written, “born of citizen parents” instead of “natural born citizen”, and everyone would agree they were following Vattel. Or they could have required the President be a NATIVE, which would probably mean they were following Vattel. But instead, they used a term that all of them were well familiar with, having used it all their lives. And the meaning of that term is found by understanding the common law MEANING of NBS.
As WKA put it:
“The Constitution nowhere defines the meaning of these words, either by way of inclusion or of exclusion, except insofar as this is done by the affirmative declaration that “all persons born or naturalized in the United States, and subject to the jurisdiction thereof, are citizens of the United States.” In this as in other respects, it must be interpreted in the light of the common law, the principles and history of which were familiarly known to the framers of the Constitution. Minor v. Happersett, 21 Wall. 162; Ex parte Wilson, 114 U.S. 417, 422; Boyd v. United States, 116 U.S. 616, 624, 625; Smith v. Alabama, 124 U.S. 465. The language of the Constitution, as has been well said, could not be understood without reference to the common law. Kent Com. 336; Bradley, J., in Moore v. United States, 91 U.S. 270, 274. [p655]
In Minor v. Happersett, Chief Justice Waite, when construing, in behalf of the court, the very provision of the Fourteenth Amendment now in question, said: “The Constitution does not, in words, say who shall be natural-born citizens. Resort must be had elsewhere to ascertain that.” And he proceeded to resort to the common law as an aid in the construction of this provision. 21 Wall. 167.
In Smith v. Alabama, Mr. Justice Matthews, delivering the judgment of the court, said:
There is no common law of the United States, in the sense of a national customary law, distinct from the common law of England as adopted by the several States each for itself, applied as its local law, and subject to such alteration as may be provided by its own statutes. . . . There is, however, one clear exception to the statement that there is no national common law. The interpretation of the Constitution of the United States is necessarily influenced by the fact that its provisions are framed in the language of the English common law, and are to be read in the light of its history.”
In this as in other respects, it must be interpreted in the light of the common law, the principles and history of which were familiarly known to the framers of the Constitution.
And he proceeded to resort to the common law as an aid in the construction of this provision.
The interpretation of the Constitution of the United States is necessarily influenced by the fact that its provisions are framed in the language of the English common law, and are to be read in the light of its history.
Principles and application. I can understand your confusion.
IMO your principles are suspect and your application of them is obvious.
As you say, "...English common law DID provide the language of law for the Founders."
Beyond that, the language of law, our common law became uniquely American.