OP, please do not co-opt the BLM movement for the purposes of justifying your sweeping generalizations about gamers.
As an African American who has to live in constant fear of the police, I find your parallels ignorant and distasteful. The reason why #AllLivesMatter, #BlueLivesMatter, and #NotAllCops are problematic is because their very similar naming conventions attempt to detract from a very specific concern as it pertains black lives, since it's not all lives that are systematically oppressed and killed by cops.
However, the problem with gaming culture is not specific to gamers. It is a problem with internet culture in general (which is a result of human behavior in general when placed in that kind of environment). Furthermore, it is perfectly reasonable for gamers to directly respond to recent social movements specifically aimed at criticizing gamers. These particular movements aren't even specific to the victims of the toxicity in gaming, but rather the opposite, the alleged culprits. In that case, it isn't a matter of detracting from the concerns of the victims like it is when people counter #BlackLivesMatter, it's actually responding to the criticism.
So essentially, what is the response to the criticism? It appears to be that gamers have had enough stigma to deal with ever since the conception of video games, and they are pushing back. Video games (and subsequently the people who play them) are becoming more and more mainstream, so much so that the demographics for gamers are too broad to draw conclusions about behavior specific to gamers. It just doesn't make sense, as gamers are diverse in both their personalities and behaviors, and there isn't even an exclusive type of gaming culture anymore.
I said all that to say this. It's fine if you have a problem with gamers pushing back on this issue, but do not equate what they're doing to people pushing back against the BLM movement because it's not the same thing and it devalues the merit of BLM activists when you say that it is.