The New Englander, Volume 3 (1845) states: "The expression ‘citizen of the United States occurs in the clauses prescribing qualifications for Representatives, for Senators, and for President. In the latter, the term ‘natural born citizen’ is used and excludes all persons owing allegiance by birth to foreign states."
The lawyer's argument wasn't that the child wasn't a natural born citizen. He was saying she wasn't a citizen at all, despite being born in the United States to a citizen Mother. I know of no contemporaneous evidence that the lawyer might have gotten that idea from.
The book I cited the quote from was, I believe, a Yale Law book. Perhaps he got the idea from books such as that? You cannot dismiss the idea of "citizen parents" as being peculiar, even the supreme court acknowledges that aspect of it.
...it was never doubted that all children born in a country of parents who were its citizens became themselves, upon their birth, citizens also. These were natives or natural-born citizens, as distinguished from aliens or foreigners. Some authorities go further and include as citizens children born within the jurisdiction without reference to the citizenship of their parents. As to this class there have been doubts, but never as to the first.