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!
Obligations on parents to generate that paper trail exist now but there are still many ways people can fall into the cracks. The US has generally been far more accommodating of Americans without documentation out of necessity than I think people realize. Some parents choose this for their children, either intentionally or through negligence, and those children need a way to bootstrap their documentation as adults.
There was a large contingent of Americans born outside the US to American parents in the aftermath of WW2 that frequently had little or no documentation.