Will you accept TV-aired miniseries? If so, how about Into the West? That miniseries followed two families, one Native American and the other white frontier settlers, and got pretty gritty?
Failing that... how about "Cold Case"? I know you say no cop shows, but of the four main detectives through most of the show's run time, three represent minorities with pretty gritty backgrounds:
--- one is the daughter of an alcoholic and grew up on welfare (white but female - incidentally the first female homicide cop)
--- one is Puerto Rican/Cubano, the son of (illegal?) immigrants who fled Castro; his brother is also a victim of sexual abuse
--- one is black and on at least one episode admits that "kids like me dreamed of growing up to become Black Diamonds [a gang]" (I'm paraphrasing here).
Oh, and in the second half or so of the show's running, they add a fifth cop - a black woman and a single mom.
True, the fourth detective and the lieutenant seems to be middle-class white; but the majority of the show aren't what I'd consider the privileged class. These characters may have "made it" in the sense that they have a steady job, but there family members are often the epitome of disfunction and social disadvantage. Actually, that show goes out of its way to show how the cases being investigated intersect with the characters' persoanl dramas, because the cases investigated are selected from the "cold cases" - ones that happened years ago. So for instance when the star detective Lily Rush decides to investigate a murder from the 1920s, it is because she empathizes with the victim's descendant who talks about messed-up family legacies; it resonates so well precisely because Lily is going through something similar with her own family.