Exactly whose "What about black on black crime" argument doesn't work? Shouldn't we have a quotation from someone making that argument? How about several quotations from several people, because maybe they are making different arguments? (Ms. Temple says "Let's talk," but then she does all the talking.)
"What about ...?" is never a serious question, so rebutting it never makes a persuasive case. Not that the rebuttal here makes any sense - the wrong stats are used, and the wrong inferences are drawn. For example, that Black cops are as likely as White cops to kill Black people is consistent with both the inference that racism is not at play and the inference that the "problem is institutional." But Ms. Temple blithely assumes the latter must be the case, because, well, it just has to be, doesn't it? That is not how statistics work.
One reason to mention Black on Black crime is to argue that Black lives apparently don't matter to Black people. That would be a problem for BLM if its name were its mission. Fortunately, Ms. Temple has disabused us of that notion. BLM's mission is not to establish that Black lives matter; it is to end White supremacy. That's a fine goal, but it's not what the name of the organization implies is its focus. So people wonder aloud why Black lives should matter to White people if they don't matter to Black people. (Ms. Temple's authority on Black on Black crime is a burglar and not a killer. I guess property matters more to Ms. Temple than Black lives do. )
At the end of the day, the question Black leaders should be asking is what actions will best improve the quality (and safety) of Black lives. IMHO, too much effort is being directed at changing White people's behavior instead of changing Black people's behavior. That's not a moral judgement; it's a tactical one, an answer to the question of how best to solve the problem that needs to be solved.
The history of Black people in the US is unique, but at some point in the march to equality, parallel development becomes a viable approach. Racism will never go away, but oppression can be overcome by achievement. White supremacists have lost the fight to keep Blacks from achieving the critical mass of capital, talent, and commitment necessary to equal achievement.
Now the Black community must recognize that it has won that particular battle and start putting those resources to work, not just by tapping them anecdotally, but by preparing ALL young Black people to access them. Right now, Black America has the reverse Wile E. Coyote problem: they need to look down and notice that there is solid ground beneath their feet.
Institutions ossify, and the Black equality "industry" has its entrenched players like any other. These people earn their living fighting their good fight, and they will contend that their fight is the right fight long after it no longer is the best fight. If skin color is all that separates the races, then we should expect them to share the same foibles, including bureaucratic defensiveness.
As a friend of mine used to say about business, assume the sale. While BLM is working to end White supremacy, Black Americans would be well advised to live AS IF it has already ended. Otherwise, it won't.