The problem with the name and the slogan “Black Lives Matter” isn’t because “all lives matter”. I agree that this is a foolish and ignorant response to the slogan, at best; and at worst, it is a deliberate attempt to deny the very real and unique jeopardy faced by African-Americans in the US.
The problem is that the organisation using the name and the slogan “Black lives matter” doesn’t mean it.
They would avoid a lot of the criticism they receive over their complete indifference to the murder and violence within the Black community if they just called themselves something like the “coalition against racist police violence”, since that is actually all they focus on.