I think you're interpreting the argument in reverse. It's not to say that the white murderer should have been killed. It's to say the black men who had committed far lesser crimes (if any crime at all) should certainly have been taken alive. (No trip to Burger King required.) That they are killed so automatically for such minor offenses indicates that it is a lie to say that all lives matter. Clearly black lives did not matter to the cops who killed them for so little reason. Meanwhile, the life of the white murderer clearly did matter to them, which is why they took the risk of apprehending an armed man instead of just shooting him on sight.

Make is make sense. Make the cops treat the black men the same as they treat the white men.

Again, your example of the white looter during the riots doesn't measure up the same because one is during a riot. That one white man's death wasn't amplified in their context, and neither were all the black deaths in that same setting that you clearly haven't even heard of at all. Even right now during these riots, lots of people of all races are being hurt/killed/blinded and you aren't hearing about them.

But BLM is specifically focused on amplifying that for too many people black lives matter less than all other lives, and that's resulting in us dying when anyone else would have been allowed to live. That's the point, so no, they aren't going to go out of their way to make other points too.