I was born before that and issued my SSN at birth.
The first pilot project to issue SSNs with the birth certificate automatically was in 1987. You can read the history here:
https://www.ssa.gov/policy/docs/ssb/v56n1/v56n1p83.pdf
Prior to that, getting the SSN required giving your birth certificate to the government. If the family wasn't getting government benefits, many didn't bother.
The first pilot project to issue SSNs with the birth certificate automatically was in 1987. You can read the history here:
https://www.ssa.gov/policy/docs/ssb/v56n1/v56n1p83.pdf
Prior to that, getting the SSN required giving your birth certificate to the government. If the family wasn't getting government benefits, many didn't bother.