4 stars; rated PG-13, 93 minutes “Won’t You Be My Neighbor?,” Morgan Neville’s admiring portrait of public television pioneer Fred Rogers, feels reverse-engineered to soothe the rapidly fraying nerves of a country mired in political and pop-culture food fights. Revisiting Rogers’ signature TV show, “Mr. Rogers’ Neighborhood,” and the kind but steadfastly enigmatic man behind it, Neville has created a film that operates both as a dewy-eyed nostalgia trip and stirring appeal for civility. In 1967, when “Won’t You Be My Neighbor?” begins, Fred Rogers had been working in local Pittsburgh television and had attended the Presbyterian seminary when he began to conceptualize children’s programming that spoke thoughtfully and usefully to the emotional needs of a young audience.