when a team wins the fans don't worry much about former players or coaches.
if the team is losing they start to blame the former coach , especially in college sports since that guy recruited the players
if things go really bad they say "maybe that last coach was not so bad after all"
Meh. That depends. Not every coach is popular with a fanbase even if they do win. And what kind of terms you leave on makes a difference too. Babcock has alienated multiple fanbases despite winning everywhere he's been. When a guy acts like an ass, people tend to wish ill of him in the future. Go figure.