Root cause of this is Great Society liberal programs. Did the Moynihan Report mention the state of black families before they were destroyed in the 60s? Cohabitation of unmarried couples had nothing to do with it.
Illegitimate kids are the offspring of unmarried people who cohabit, however briefly. Then as now, the cohabiting was not all that brief but not long enough for rearing the resulting kids.
I recommended earlier that self-described conservative fans of shacking up should read Myron Magnet's The Dream and the Nightmare: The Sixties' Legacy to the Underclass, a book President Bush greatly admires. I recommend it again.
It's probably something of a symbiotic relationship. The Great Society hurt society. The sexual revolution hurt society. Both hitting at the same time was a major catastrophe.
In addition, they feed off one another. The Great Society gave men an excuse to father kids and not worry about them. The sexual revolution produced more illegitimate kids in need of government programs. Even the upscale working women who have kids out of wedlock eventually turn to government for assorted benefits. They want day care subsidies, more community centers for their unattended kids to hang out in, and so on. Look at the voting patterns of single women. They vote Democrat by a HUGE margin.