I'll say I think you're being a little harsh on Allen. In the grand scheme of things goals and assists are not his game and throw in a few pointless goals and assists, then you couldn't make that accusation, but would he then be a success? I think its just a throw away stat so the wider point doesn't need to be evidenced. His job is not be be shooting or making the kind of passes that lead to goal, so why is the lack of him doing that relevant?
I think he's just redundant in your set up, but I would say when he arrived that wasn't the case, got player of the month when he first arrived did he? When you've got players like Lucas, or Gerrard playing a Pirlo role from deep, or just are playing very fast fluid counter attacking play, his role (as such he as a player) is just a spare part.
Joe Allen was always of the quality he was going to be a squad player, but you then take a specialised squad player and play him outside of the role and wonder why he looks ****. He probably shouldn't be in there, but I don't think you can aim any vitriol at him personally. If Rodgers brought him in, he has to do so with the knowledge Liverpool are going to be playing a system which suits him and has a place for him.