It wasn't Johnson's Great Society, it was feminism.
Specifically, radical white feminism, in which white women drew from the power they already possessed as benefits of the white patriarchy (even as they railed against that patriarchy). Their goal was to destroy the family...and the black family was already weak as a result of slavery and Jim Crow.
Ironically, most white women didn't swallow the "strong, independent woman who doesn't need a man" rhetoric (as noted by the fact that most white women voted for Donald Trump), but black women did as a result of always having been in the workforce and never having been on the "wife and mother" pedestal....and that was a product of slavery and Jim Crow. Black women did not get the benefits of patriarchy that white women got. Black women were easy marks for the "strong, independent woman who doesn't need a man" rhetoric.
White radical feminists, drawing from the power of the white patriarchy (that they despised) could offer perks to black women that mere black liberation could not offer. But the price was the black family.
Regarding the Great Society, most blame is placed on the Aid to Dependent Children (ADC) program and its "no man in the house" provision. However, that's greatly overblown. ADC was actually instituted in 1939, during the Great Depression, with the intention of keeping white mothers out of the workforce and out of competition with white men. At the time, white mothers received payments even if their unemployed husbands were in the house. I point out "white" here because black women were prohibited from receiving ADC...because they were already considered part of the workforce (they had no benefit from the patriarchy).
After the Civil Rights Act, black women began receiving ADC as white women did. However, at the same time, several states (it was a state-administered program) instituted the "no man in the house" rule. The Supreme Court outlawed that rule in 1969. So, "no man in the house" had only been in effect for five years and not in all states. It was a minor factor, limited in time and area, in the dissolution of the black family. Radical feminism has been the real cause of the situation we are in today, and the reason why the black family could not "get it together" after the Civil Rights Era began to relieve us of racist pressures.