LEt me clarify one thing you said.
Birth records that do not list SSNd are not considered individually-identifiable health information, but they are considered protected information covered under the 1974 Privacy Act.
For years, they were open records, long before FOIA, etc. When you got a typical “copy of your birth certificate,” it was a negative photostatic copy of a certificate that had been completed by a hospital, or in the case of a home birth, the attending MD, RN or non-nurse midwife. That certificate was then sent to the county or state to be kept into some master book/file, and you could get a photocopy.
But then it was determined that certificate had too much private information, race of parents, were parents married, had mother been tested for syphilis, etc?
So rather than give you photo copy of original, you now get a certificate, which is really an abstract of the original information as submitted by doctor, hospital, etc. minus the medical data.
This was long before newborns got SSANs at birth. In mid 20th century, you got SSAN when you went to SS Administration office so you could get first “real job”
Some jurisdictions still only give the “certificate or birth record” to the parents or child involved.
I am not aware of any case law about attempts by a non-party to obtain information under FOIA request. Way out of my area.
Good luck.