I see two sides on this.
I think it must be acknowledged and emphasized that this is happening to Black people much more than other groups, and that they are being specifically targeted. I also think it is easier to get a buy-in about police brutality from society at large if we talk about maltreatment by police officers in a general sense.
If a person can identify with the person being mistreated, they are more likely to empathize and want action taken. Right now, some people have sufficiently distanced themselves by believing most Black people who are mistreated or killed by officers were criminals not worthy of being treated fairly or humanely. They don’t believe that it would ever happen to them, they don’t feel at risk themselves, and they don’t care that other people are at risk.
So, for public support, I think it might be worthwhile to broaden the scope, but in terms of training officers, they need diversity training, deprogramming, and to be screened better before they are even accepted into the force.