The company, the University, produces a product, the football team. The fans, the customers, pay for the product. When the product becomes less desirable to the the customers, the fans, many stop purchasing the product, the tickets to see the football team, then the company, the university, loses money and if something doesn't change, the company, the University, goes belly up.
Fans do influence coaching hires and fires. I've always maintained, especially when it comes to UT sports and football in particular, that all decisions to fire are mainly made on a spreadsheet. The coach will be fired when it's financially responsible to fire him. When the losses outweigh the cost of firing and hiring a new coach.