https://www.politifact.com/article/2020/aug/28/ask-politifact-does-black-lives-matter-aim-destroy/
- Critics of Black Lives Matter point to some of the statements on its website, including that it says "we disrupt the Western-prescribed nuclear family structure.”
- Black Lives Matter’s full statements about family show that it wants people to support one another broadly beyond the nuclear family.
- Black Lives Matter also calls for family-friendly spaces and equal rights for mothers.
...
An attack made less often is that Black Lives Matter wants to abolish the traditional family.
For example, at the Republican National Convention, former NFL player Jack Brewer said the organization "openly on their website calls for the destruction of the nuclear family. My fellow Americans, our families need each other. We need black fathers in the homes with their wives and children."
We found that while Black Lives Matter seeks change in how "family" is defined, especially with respect to public policy, it’s a leap to conclude that it wants to eliminate traditional family structures.
I agree with politifact's interpretation of what they mean their goals to be.
Often (guessing more than average and more than white families), black families are "non-nuclear". So they are just saying they want more acceptance and support of that, rather than trying to push the "nuclear" family as the ideal and accepted.
I see nothing wrong with that, and nothing about that is wanting to "destroy" the nuclear family.