Nottingham Forest are a very good example of what difference a good coach can have on a club in disarray. They had mad unrest between players and fans, disliked directors, questionable recruitment, 5 wins in 27 before Hughton sacked. Now complete harmony, just below play-offs, just 3 defeats since he took over.
Meanwhile... sacking Benitez won't magically solve all of our problems, but you can't discount the potential effect that could be had by swapping him with a genuinely good manager or even just a mediocre one that the fans and players can have faith in.