Some of the founders of BLM are self-proclaimed Marxists. I don't think it is much of a stretch to say there is some sort of a connection between Marxism and being anti-family.
This refers to ways our society benefits 2-parent households and makes things harder for single parents. They suggest addressing this by providing more support and community for each other, not by tearing down nuclear families.
As an analogy, installing wheelchair ramps doesn't prevent people from climbing stairs, it just makes building more accessible.
I see no way to read that as BLM supports the nuclear family. So the largest social justice movement in the US doesn't support two parent families. That seems pretty significant
Again, why in the world would it be necessary to disrupt the nuclear family to gain greater connection to extended family and the community??
Which in a rural white community would literally be a Sunday church sermon.