You would think it's that easy as the "government knows all". But from working on government databases and systems, it's hardly the case. Disparate systems, competing departments, endless workshops that hold up progress, political spats, etc. And this isn't even in a country like the US that has separate states!
The data is misused in just about every way. From colleges using it as a student id number to people committing crimes with it.
https://www.ssa.gov/pubs/EN-05-10096.pdf
USCIS lists as common proofs of citizenship: US birth certificate, US passport, certificate of citizenship, or naturalization certificate.
https://www.uscis.gov/sites/default/files/USCIS/Resources/A4...
https://www.uscis.gov/i-9-central/acceptable-documents
(an unrestricted SSN is proof of eligibility to work in the US)
Sometimes it's better to be paid than right. (Especially on your first few jobs)