Is your argument that a genuine, good faith interpretation of "Black Lives Matter" is "Only Black Lives Matter"?
This isn't how English works. If I say "I like your mom" to an SO, they wouldn't interpret it as I don't like them and instead like their mom. I don't have to say "I like your mom too".
Yes, genocides are emotional. Watching children being blown up is something that should upset you. That's actually happening in the real world.
Emotion isn't the only thing that should inform your decisions, but pretending like you shouldn't be upset at watching kids being blown up, or begging for their parents, or whatever else have you is just foolish.