Actually, the core problem is that when (usually) white, middle-class America sees a police boot on someone's face their instinct is to say, "Well, I guess he deserved it." This is particularly the case for working middle-class mothers. Based on anecdotal conversations, when they see a story about the police shooting an unarmed man in the back, they think, "Well, the guy was probably going to grab someone's little girl and take her hostage." When confronted with stories where the cop was undeniably wrong, they say, "Oh just a bad apple."
Another problem is when (usually) black America sees a police boot on someone's face, and notes that statistically it's usually a black face, and their instinct is to say, "Well, I guess it was because he was black." When confronted with stories where the face is white, they say nothing.
This breaks my heart. We should never allow a police boot on anyone's face for any reason. The police are public servants, our servants, there to keep the peace, and stop violence from happening. They are not the punishers - that is the role of the court. They are not above the law - they need to hold themselves to a higher standard of conduct. The police should be the best of us, but instead we get low-to-average intelligence physically large men who can keep themselves clean and fit, fill out basic paperwork, and who can follow orders. Emboldened by the attitudes of white, middle-class women that "cracking heads" is a sign of order, also a sign of machismo, and absent any authority to stop it, it continues, and it gets worse.
Now, cops go out on a disturbance call, escalate it to violence, and then they 'hurt the bad guys' - cracks some heads, throw 'em in jail. The cop moves on, but 'the bad guys' keep getting hurt, in local lock up, by a justice system that will never hear their case, and then for years by a privatized penal system that takes every economic and social advantage of the prisoner that it legally can.
What we have now is a police force whose individual instinct and unmet need for "respect" (actually, dominance) drives their decisions. No-one can stop them. The police have each other's backs. The judiciary has the backs of the police. And the voters are split between two very different, very wrong reactions to the problem - which takes back seat to things like "the economy" and "abortion" in every election.
Right now, the best we can do is tell the stories of the people who've been harmed by the police. We need to focus on white, middle-class people, to address both mistaken instincts. We need to fund independent police-malpractice commissions with real teeth, to investigate allegations of wrong-doing. More than anything, we need to reform the way police work is done, making it illegal, for example, for a uniformed cop to coerce a suspect, to lie about the law, or to escalate a situation. We need a "broken windows" policy applied to the police where even small breaches of policy - words of disrespect, for example - trigger a reprimand. We need to ask ourselves why cops swarm on every encounter with a citizen, why they have their hand on their gun when they walk up to your car, why they beat people up so much and get away with it.