"Alabama's football program may survive detached from the school somehow, but would water polo? What about a football program at a non-powerhouse but still FBS school like the University of Akron?"
At a big time sports school like Alabama, football pays the budget for all varsity sports. This is the case at a lot of schools, not just big time programs. At my alma mater, Siena, basketball was the cash cow that paid the bills for all sports programs. At a school like Akron, they make their athletic dept. money off games with big programs early in the season - "cupcake" games. Alabama will pay a shitty program to travel to their home field and get trounced by 70

in exchange for $500-600K. The arrangement is mutually beneficial - 'Bama gets an easy win, Akron gets to field sports programs, and once in a while there's an upset that garners a lot of attention for the small school (Appalachian State, for example).