Both situations bother me. I understand the sentiment of not losing sleep over someone who has had a highly checkered past and losing sleep over a guy who didn't. But this isn't a scenario in which we are lost at sea on a raft and we have to choose one person to toss overboard or we'll all perish. We don't have to choose either. Indeed, I'd argue that we should not make any such distinctions, no matter how sympathetic we might be by one's history over the other.
We are a country of laws, and when those laws are not applied evenly, we fail as a society to live up to our ideals. We see the same kind of garbage throughout our politics today, and everyone complains vehemently about it -- as well they should. To me, the comparison of these two incidents is about the application of law, not the victims. If we shrug one incident off because we don't like the past behavior of one of the men who was killed by police, we do a disservice to the concept of equal protection. And if we don't have that, where does that leave us? Identifying certain groups who are above the law, and others who deserve harsher administration of law? That can't lead to a good place.
I am not making any judgement your "humanity," but the way I see it is that indifference to one person who was unjustly killed will make it easier to shrug off the next incident, and the next. And if it mostly happens to people who we don't really care about, it becomes normal ... even accepted. I see it as a significant factor that contributes to, as you put it, the "kind of shit that is tearing this country apart."
We should care deeply about both -- not because they were both upright, model citizens, but because they are both citizens, period. What happened to these men is tragic and terrible, and what it says about our society that this sort of thing happens with such frequency is just as tragic and terrible.