Sometimes firing the coach helps, sometimes it doesn’t. Cleveland is too bad a team for a new coach to make them a Super Bowl contender.
Certainly, teams have won championships with mediocre coaches. There are circumstances, though, where a new coach is exactly what a team needs, especially when the old coach has been there a long time and has become blind to the team’s weaknesses.
I think that football coaches affect the game more than just about any other coach. He calls all the plays and defensive alignments, as well as giving the team specific practice for each opponent.
As well, the front office is important because there is so much player turnover due to the hard salary cap. All the teams have the same budget so you can’t just pick up any old free agent to fill a void.