> We have plenty of naturalized immigrants with no traceable documentation of their birth because they grew up in the middle of nowhere.
Sure, because the documentation required used to be less.
When I was first getting my driver's license, you could establish your identity by having a birth certificate or some sort of vouching process. These days, you need a birth certificate.
It would be hard to immigrate today without a birth certificate, in part because it would be hard to get a passport without it, and it's hard to enter the country without a passport. Countries where record keeping was historically poor tend to have more accessible ways to obtain a passport without a birth certificate... A passport is sufficient proof of birth in an immigration file, and a certificate of naturalization is sufficient proof of citizenship, so a birth certificate isn't explicitly necessary.