For reference, the US has procedures for this: https://www.usa.gov/citizenship-no-birth-certificate because people without birth certificates are still somewhat common, even children.
Vermont didn't require it until 1955!
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Vermont didn't require it until 1955!
IIRC, you need a couple of people to sign affidavits that affirm you are who you say you are. That's the start of the "paper trail" and then you start rebuilding your document pool.
Getting married and changing your last name is similar (although with fewer documents etc).
It is more involved than just affidavits. The US uses databases on every citizen, some not formally acquired, that can be used to "duck type" individual identity. An affidavit is primarily used to bootstrap the entity resolution process. With only a couple touch points they can reconstruct identity with high probability. It may feel like a "trust me bro" process but it really isn't.
It is related to how the provided information on credit applications is not used to inform the creditor. They already have access to all of this information and are more interested in if your representation matches what they already know.