Most US citizens couldn't prove they are citizens, at least without a fingers-crossed records search IF they can remember the county they were born in. Stats say only around 10% of americans could easily put their hands on their birth certificate. Almost no one can produce one at a checkpoint if demanded, and its rare for people to even have one in their possession at home.
A passport proves citizenship, but its absence doesn't disprove it.
Voting cards and social security cards aren't identification. State issued cards like drivers licenses, state ID cards or even realID cards do not prove federal citizenship (although they do prove identity).
Sources: https://www.brennancenter.org/our-work/analysis-opinion/mill...
https://en.wikipedia.org/wiki/Identity_documents_in_the_Unit...
Of course you are right, basically no one carries their birth certificate around. Which is probably countered by the fact that birth certificates are pretty easily falsifiable because there is no standardization of them.
