I looked this up and while your first sentence is true, the second (non-parenthetical) sentence is only true if you did not require any of the other services that required a SSN. There's a list of those under "Exhibit 2" (about 2/3 of the way down the page) on the SSA's website:
tl;dr: If you had a bank account, applied for a federal benefit, were on food stamps, applied for school lunch, or did any number of other financial or government transactions, you needed a SSN starting in the 1970s. That's enough of an incentive that many parents might've just applied at birth, figuring that their kid will eventually need it. Also everyone born 1968-1981 would've likely gotten one in 1986, when the change you mentioned about dependents was enacted, and then after 1988 they started being required for issuance of a birth certificate.
I stand corrected. Thanks. I didn't bother to look it up, since I'm old and got mine when I started working. Although people born 1968-1981 were getting SSNs where they currently lived, which is not necessarily where they were born; which was the original point.
https://www.ssa.gov/policy/docs/ssb/v69n2/v69n2p55.html
tl;dr: If you had a bank account, applied for a federal benefit, were on food stamps, applied for school lunch, or did any number of other financial or government transactions, you needed a SSN starting in the 1970s. That's enough of an incentive that many parents might've just applied at birth, figuring that their kid will eventually need it. Also everyone born 1968-1981 would've likely gotten one in 1986, when the change you mentioned about dependents was enacted, and then after 1988 they started being required for issuance of a birth certificate.