Years ago I ordered a certified copy of my long form.
It came back and what it was/is is a printout on photo-sensitive paper of a copy of my original BC from microfiche.
It had the original info about where I was born, info about my parents, the doctors signature, the hospital registrars signature. As well as initials of someone from the Dept of Vital Stats from NYS, and the State seal impressed.
So there may be no “paper” form, per se. But odds are very high that somewhere, there is an image file of sorts.
A single reel of microfiche could hold thousands, perhaps tens of thousands, of images.
See my post #37
bttt