SOUTH BEND, Ind. — It was more than an hour into Thursday’s Democratic debate before the community activists watching back home finally heard Mayor Pete Buttigieg face the tough questions they were waiting for.

Why was the South Bend police department just 6 percent black in a city that is 26 percent African-American? How had a white officer fatally shot a black man in a routine encounter 11 days earlier?

And why, after years of policing traumas like the one in Ferguson, Mo., was it taking so long to bring accountability to South Bend?

Mr. Buttigieg admitted to falling short. “I couldn’t get it done,” he said about failing to increase diversity on the police force. He expressed anguish for the family of the dead man, Eric Logan.