That doesn't make sense. Whether good or bad, he does direct coaches, approve game plans, decide when and how long to practice, participate in the draft process, make decisions during the game such as when to challenge, when to go for 2, to kick a FG or go for it etc, play a part in personnel decisions .... Obviously he has a role that influences how things go. Irrelevant is the water boy or locker room attendant.
Besides, to say he is irrelevant is to put responsibility on him when they lose because if a head coach is irrelevant he is hurting the team.
As for one playoff run undoing 9 years, I never said it did or would, and in fact said that whatever consideration it would get him would be short lived if it turned out to be a 1 year anomaly.