It’s not the only reason, just the first and easiest. Also, they can say they’re not citizens all day, but unless they legally revoke it (expensive and difficult without dual citizenship because no one wants to deal with a stateless person), the Department of State knows they are. They think that they’re citizens of the state in which they were born, and they want the passport American Samoans get, which is for American nationals, not citizens. They pretty much take real things that exist and try to twist them into whatever alternate reality they’re living in.
As for the birth certificate, I’m 99% sure what that means is that they requested an apostille. It’s basically a certificate signed by the (state level) secretary of state saying the document is authentic that’s attached to certain legal documents. It’s required by certain countries, so I’m assuming they said they need it for a non-Hague country and they think that does something a lot more interesting than just saying it’s a real birth certificate.