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!
When McCain was running for president, there was a big court case about whether being born in the Canal Zone (a U.S. territory) qualified as being a "natural born citizen". And I made the connection - "Wait. The Philippines was a U.S. territory in 1939. Shouldn't dad have had birthright citizenship?"
Moot point by then, he'd already been a citizen for ~40 years, and died the next year. But it was wild to think that the 10+ years of immigration hassles were basically due to an administrative fuck-up, and that legally, he should have had citizenship all along. The process you link wouldn't work for him, either, because the Philippines is not a U.S. territory now.