I won't argue that the motto isn't exclusive to some degree. Everything in life is, even if they aren't logically so. For example, you could love Picasso more than Van Gogh, but still love the latter. If you're a museum curator, though, you only have limited space...the fact that you choose to pick Picasso artworks doesn't mean that Van Gogh is terrible, and yet, you had to prioritize.
The BLM group don't hate non-blacks. But they believe the disenfranchisement of blacks needs to be re-prioritized. While it's not a zero-sum game, sure, in the short term, tradeoffs are made...if next year's recruitment is 20% black...that 18% had to out of some other demographic.
So, in my rosy-lens world, the two groups aren't in conflict, but it's realistic to recognize the tradeoffs. But what isn't acceptable, in my opinion, is for either group to make this into a false-dichotomy...that's why I like Zuckerberg's message...it's not an easy topic, but just crossing shit out escalates discord in a way that is not warranted, as if the people writing "Black Lives Matter" were just one step away from drawing swastikas.
To your point about poor white people being just as disenfranchised as the average black person...I don't disagree with that. I grew up in the Midwest to refugee parents and in my lesser moments was annoyed that I got grouped in with the affluent Chinese and Korean families, as if all Asians had equal opportunity to get great grades and live out the stereotype of the golden minority. On the other hand, I realize that while I may not have started off financially well, there were far fewer barriers for me in terms of other people throwing up obstacles...and that was probably because the Asian community, overall, has positive stereotypes associated with it (in terms of general society...obviously not so much in other areas). This was not the case for the few blacks I knew. You could come from a well-off family and still be followed around in stores or targeted for a stop-and-frisk. I'm not sure growing up middle-class but in what might feel like a surveillance state is better than growing up poor, yet encouraged.
So I wouldn't say the BLM people don't care about the other kinds of disenfranchisement. They just believe they need to focus on one kind to be successful. Think about Occupy Wall Street...a lot of their general ideas were things many people would like, but because its organizers didn't want to be exclusive, that meant casting a very wide umbrella and attracting people whose goals did not gel well with OWS, making the group seem very diffuse. And so OWS was mocked for being weak and daydreamy.
The most successful groups are the focused groups. Ask the unions and the gun rights groups.