I've marched in protest over a police killing. It was not about race, it was a SWAT raid that was a forfeiture fishing trip. I care about the police unjustifiably killing people. They kill people of every race. Before you bring up per-capita, women are 51% of the population but the number of women killed by the police is the most disproportionate statistical ratio of all. Why?
The number of black people killed by the police is a pimple on an elephant's ass compared to the number not killed by the police so yes, it matters. But when it comes to issues of race I see a larger issue in racism itself. The root of racism is to have a low opinion of a group based upon their race. Without that, there is no racism.
Of course the majority of black people care about black on black crime and murder. If you Google "say his name" you will find names of people not killed by the police. Try "Mekhi James." But it is drowned out by the vast number of police oriented ones from the BLM crowd. This gives an impression that black on black killing is a low priority. People's perceptions are their reality so that matters and it's why they bring it up.
That leads to nobody being surprised by violence, arson, looting and murder as a method of protest to honor the memory of the fallen by black people. That's as low an expectation of a racial group as it gets and it perpetuates the racism of low expectations of black people. How will racism based upon low expectations of black people ever go away in the face of that? If that is letting the police off the hook you could be accused of letting black killers off the hook. I doubt that either of us have those intentions. At least I don't.
You mentioned the inability of the black community to make black on black violence go away. Why do you think that is? No community has made it all go away so that is not the issue. It exists in a large enough proportion compared to other demographics to once again, give an impression that it is business as usual/expected behavior for black people. That leaves people with low expectations/impressions of black people as a group. That is racism of course. But what is to be done about it? Complain about the police and white privilege? Why do you think that the black community has that disproportionately large problem? What is being done by the community to resolve it? That is an honest question.