"All lives matter" as a rebuttal or reminder is about as useful as "LOL it's not actually free you forgot about taxes" when someone mentions "free" healthcare. Everyone already knew that. Context is always part of understanding language and in both those cases the context is pretty damn clear—these aren't even ambiguous or difficult ones. It's just being intentionally obtuse for a plainly-bullshit "gotcha".

Are you implying you didn't already know that "black lives matter"? I'm not following that part of the arguement

No, it's that the message of "black lives matter" is intended as a reminder to a society that acts like it has forgotten this or doesn't give it enough consideration. "All lives matter" as a rebuttal to it is a deliberately-obtuse "gotcha". "Well yeah but all lives matter, so there, you should use that slogan instead of making it about race". Everyone got that already. What (apparently) needed to be pointed out was a more specific thing. And most people understand context or they wouldn't be able to communicate, period, and the context of "black lives matter" obviously doesn't mean other lives don't, as anyone who's not trying to play jerkass quibbling games would already recognize. It's a deliberate misunderstanding for the sake of distraction.

Logically if people have forgotten black lives matter then they have forgotten that all lives matter. I think people who get caught up arguing against logic are purposefully being devisive.

In the context of police brutality, white lives don't matter. Who protests when an unarmed white person is killed by police?

Well the point of saying "All Lives Matter" isn't actually to claim that all lives matter, so a "good faith" way would kind of be missing the point. The people who invented it and say it don't mean the lives of poor black kids in the inner cities, families of Salvadoran immigrants apprehended at the border, transgender activists in San Francisco, etc... the phrase "All Lives Matter" is just the 2020 version of... (I'll let Lee Atwater do the rest):

You start out in 1954 by saying, “Ni\\er, ni\\er, ni\\er.” By 1968 you can’t say “ni\\er”—that hurts you, backfires. So you say stuff like, uh, forced busing, states’ rights, and all that stuff, and you’re getting so abstract. Now, you’re talking about cutting taxes, and all these things you’re talking about are totally economic things and a byproduct of them is, blacks get hurt worse than whites.… “We want to cut this,” is much more abstract than even the busing thing, uh, and a hell of a lot more abstract than “Ni\\er, ni\\er.”

This quote is essentially saying... that the root of fiscal conservatism and small government conservatism is racism. In the 1960s and still today, people support states' rights as an indirect way to hurt black people...?

This quote is terrible and one of the most racist things I've ever read. It tries to interject race into forms of political theory that have nothing to do with race. Just absolutely toxic for society.