Quoting from the Social Security office: "Must my child have a Social Security number? No. Getting a Social Security number for your newborn is voluntary. But, it is a good idea to get a number when your child is born. You can apply for a Social Security number for your baby when you apply for your baby’s birth certificate."

A couple of decades back, the law was changed to require the SSN for any claimed dependents. Most people, of course, want that ~ $4,000 deduction, so hospitals often distribute the birth registration form.

For what it's worth, I was also born in the 1970s, but didn't get my SSN until jr. high, as I recall.