The biggest issue is that the birth certificate is typically done at a very local level, usually the county, and not anyone can file one. It's often the case that a particular person at a hospital or a registered midwife needs to file the application. Parents can report a live birth outside of an institution but you need to physically go to a courthouse with the baby.
A different group of people is responsible for social security cards.

If one county clerk files a huge number of birth certificates and uses them to back social security card requests, it'll be noticed.

There's always a way around the paper trail, usually by just making sure no one bothers to look at it, but they all involve adding more people to the conspiracy, which adds more risk.
No one will notice that one doctor delivered 500 babies in one day if no one looks at the paperwork, but each person involved increases the likelihood of a mistake causing people to look, which almost certainly will cause those details to be noticed.

It's similar to how people do a huge amount of any fraud, and then once a thread of detail gets noticed the entire thing is unraveled.

Your best bet is to minimize the number of forged documents. I would predict that a single person could most easily get a non-citizen us national passport for someone to assume an American somoan identity. Since there's comparatively few non-citizen us nationals, a passport is the federally preferred method of identification. Since the territory is an edge case, there's more room for slipups, and since you're not posing as a citizen, you have an excuse not to have some records.