I'm not ready to say Allen is the full problem just yet. The defense is making plays, they are better than last year at creating turnovers, and even when they get exposed they make quick corrections. 50% of the team is working fine and ST's has been no slouch.
Still, it's the HC's responsibility to make sure everything is rolling on ST's, defense, and offense and so it is somewhat his responsiblity.
Carr spoke about details, a lot and players are having to call players out more than we've done so in the past. We can go back and forth on if Pete is bad as a play-caller and it has warrant but what seems to be more obvious than that is, the players don't respect him enough to not Fork* up on a down to down basis. There was a fear factor under Payton but more than that he was constantly teaching. During practices he'd harp about spacing among other things. Those are things that Pete cannot duplicate as a coach. He doesn't appear to be a stickler for those things, and unfortunately there's too much weight being put on the locker-room to correct those issues.
Allen has to get someone in and I don't think promoting from within is going to do it. It must be a veteran coach that commands respect, it's got to be Gruden.
Bonus points if you keep Pete on the staff. No he's not a play-caller by trade, but he didn't last that long under Payton without having value...it's just the current role doesn't fit him.