They affirm the information is "On Record", which is a distinctly different thing than affirming it is accurate.
The lies on *MY* birth certificate are also "On Record", but they are still not true.
This point is answered in my Post #233 to Red Steel.
Your grasp of public records operations is no better than his. A custodian 50 or more years after the fact has no way to verify the circumstances of a document's creation or to verify its accuracy. Yet, such information on the public records is routinely accepted (in court and in other venues) based solely on the attestation of what the records say and without any further verification of accuracy.