There are certainly off field reasons for firing the HC. We all see a LOT of programs fire a fairly successful guy and then go through a merry-go-round of coaches either not given enough time or simply worse as HCs - Texas, Tennessee, USC, Eastern Michigan, Florida, ... Finding the right balance is a trick when you have Deep Pocket Donors calling you at 2AM every night.
What if Ohio State had kept Fickell? My guess is he'd be doing pretty well, maybe not as well as Day is, but 10+ wins a year. Luke seems to be a good coach at least at a lesser program. What if UGA had kept Richt? My guess is they'd be averaging 10+ wins a season, probably not being a playoff team more than once, maybe. Richt didn't do much at Miami.