Sorry for the rambling, but hopefully this resonates. I'm sure someone has already come to this conclusion, but I feel it's worthy to re-emphasize:

Why does left insist on saying Black Lives Matter? Why do they react negatively to someone saying all lives matter, blue lives matter, etc?

The answer is for a couple of reasons:

(1) The need for moral relativism to drive division. The BLM term is more about driving an emotional response by asleep blacks that prevents reasoned discussion on proposing fixes or compromises. It also discourages meaningful communication by whites (both asleep and wake) on the issue due to frustration or fear of being "racist" If we don't talk about it, the division is insurmountable.

(2) Allowing the use of "All Lives Matter" forces a reframe of the entire issue, one that utterly destroys their narrative and provides significant opportunities for common ground. It makes it a discussion of morality. Anything that allows those asleep to renew their surface-level thoughts and notions to an immutable sense of right and wrong threatens all the enemy has done to make us rudderless ships they can push around. Is every life important? If we agree that it is, how might that allow us to bridge the ideological gap?

If it is successfully framed as a moral issue, it renders the whole of liberal orthodoxy completely untenable. The video of a Black conservative highlighting the left's silence on black vs black crime and the killing black babies in abortions is a perfect example.

So how do conservatives and conservative politicians counter this? By doing exactly what Trump, Sen. Paul, and others have done: boldly proposing legislation like the First Step act, the Justice for Breonna Taylor Act, etc. It shows those asleep conservatives care And are attempting real changes instead of fruitless expressions of emotion.

How does the average one of us help to counter the division? By showing grace and love. Darkness to light.