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.
“As you say, “...English common law DID provide the language of law for the Founders.”
Correct. So if you want to know what is meant by a legal term they used, see what it had meant all of their lives under English common law. In this case, it means that a NBC can have alien parents, if born in the USA.
As indicated by your first reply which I linked to you seem unable to grasp the concept that Vattel was adopted into American common law as you believe that it merely covers international law. On that I disagree.
@US law has never followed Vattel on citizenship, nor would Vattel have expected it to. Vattel wrote on international law, and made the point that what he was writing didnt hold true in 1758 England.
Did Vattel not apply to England since England already had a body of law?
The common law of America is uniquely American and the Founding Fathers did indeed use Vattel to form our common law.