We are all “subject” to the jurisdiction of the United States of America; we prefer the term “citizen”, but the use of the term “subject” was not unknown; as the decision in Kim Wong Ark shows.
Here is the quote that Denofrio mangles in its original language, clearly it says that the child of a foreign national born here is a “natural born subject” of the United States.
“strong enough to make a natural subject, for if he hath issue here, that issue is a natural-born subject;” and his child, as said by Mr. Binney in his essay before quoted, “if born in the country, is as much a citizen as the natural-born child of a citizen, and by operation of the same principle.”
Now you are reduced to arguing that “natural born subject” somehow doesn’t mean “natural born citizen”.