And yet the musical version of “The Outsiders” that opened on Thursday has been made with so much love and sincerity it survives with most of its heart intact. Youth is key to that survival; the cast, if not actually teenage — their singing is way too professional for that — is still credibly fresh-faced. (Five of the nine principals are making their Broadway debuts.) That there is no cynical distance between them and their characters is in itself refreshing to see. Also key to the show’s power is the director Danya Taymor’s rivetingly sensorial approach to the storytelling, even if it sometimes comes at a cost to the story itself. Many stunning things are happening on the stage of the Bernard B. Jacobs Theater — and from the sobs I heard the other night, in the audience, too.