Depends on how far you take it.
"Your kids are on their way to becoming sociopathic gang members, but far be it from me to lift a finger to help you. My kids are fine." Yeah, that's going too far on the individualistic side. Help your neighbors and your community, not just your own family.
"All children will be taken from their parent(s) at birth, and raised exclusively by the community/state." That's too far in the other direction. Take care of your own family first, don't depend on others to do it.
(And if you think I contradict myself: I'm saying take care of your own family, and help others outside it, and (to the extent you can) don't depend on others to do the same for you. Give more than you get. Yes, it's unfair. Don't look at life in a purely selfing manner.)
I can't tell, from the BLM statement, how far they want to go away from a pure no-community, nuclear-family-only position. (I must say, though, that "We dismantle the patriarchal practice that requires mothers to work 'double shifts' so that they can mother in private even as they participate in public justice work" sounds to me like "We're too busy marching in the streets to take care of our own kids, so someone should do that for us".)