If people are going as far back as Niedermayer, than I would even say the loss of Scott Stevens' to concussion problems was the beginning of the end. I don't know that we really replaced that leadership until Parise was captain, although the team had some good years with Langenbrunner wearing the C.
Who knows if Stevens got out at just the right time before the new rules were in effect. But I would have loved to see him in 05-06 when I feel like we squandered our best chance to win since '03 (excluding '12 of course). Remember how red-hot the Devils were going into that Carolina series? Elias alone had more goals than the entire Rangers team in the first round. We just choked against the Canes. Would have loved to have Stevens in that series. Niedermayer as well. But he never seemed to have enough intensity to replace Stevens as captain.
More recently though I would say the loss of Parise prevented us from continuing as a relevant team in the NHL. I don't think we'd be contenders with him on the ice, but definitely a playoff team.