You are correct. That point is why so many of us have argued for a House request that HI release all O birth documents in its possession. Anything produced by HI that reveals a non-U.S birth is game-over.
Almost certainly, a Senate trial based solely on a non-citizen father would end up in the USSC. Dicta in USSC decisions, contemporaneous writings and understood meanings at the time and history in general, suggests a lack of NBC. It seems unlikely the court could find NBC with one born to a non-citizen father.
I looks like they were trying to cover a lot of bases with those phony newpaper birth announcements. Such as “Mr. and Mrs. Obama”.