I don't know if this is the right place to have this conversation but I can't help myself. Mods - feel free to kill this.

Disclaimer, I'm not American. I don't have a dog in this fight one way or another, but I can pattern-match.

People object to "All Lives Matter" because it derails the discussion and implies that it's somehow unfair to focus, as you said, on "Whichever Lives Are Most Affected By Police Brutality At The Moment" - which in America means black people specifically. It's the same reason people object when a discussion of sexual violence is cluttered up with comments insisting that everyone recognize "women can commit rape too!" or when a discussion of social discrimination faced by disabled people meets a response like "able-bodied people can be bullied too! I was bullied for being ginger!". I've seen that kind of "what about me" response in a dozen different forms and it's almost never useful. It's a cry of not fair along the same lines as "Why can't we have a Straight Pride Parade?" "Why isn't there a White History Month?" and so on and so on.

Nobody was tweeting "All Lives Matter" before "Black Lives Matter". It's not the slogan of any particular group or movement. It's a response, and a clear implied criticism. While I wouldn't go so far as to say it's inherently racist, I'm not surprised in the least to see that motive attributed to it. If I was American I'd certainly be objecting to it too.