The powerful organizations you mentioned need to allow others to criticize them (punching up) while protecting minority groups from harm (punching down).

To your specific point, when someone says “white lives matter” or “all lives matter” they’re trying to say that those of us saying “black lives matter” are being biased, when we’re not. Black lives have institutionally not mattered as much as white lives for 400 years. So we need to fix that imbalance.

Saying “all lives matters” attempts to persuade others perhaps that institutional racism is imaginary or insignificant. As members of a privileged majority group, they are “punching down” against a marginalized group and using their collective power to further oppress. That itself proves that racism is real.

So what “all lives matter” really says is “I’m both denying and exemplifying racism in three words.”