According to a Chronicle of Higher Ed. study, the presidents at the big sports schools spend less than 3% of their time on AD stuff, even though the budgets for ADs often represent between 10-15% of the entire budget of the school. Most of the time spent by presidents is on the entire school budget. 50%+ of their time is spent literally moving around beans to see whether they can plug holes here and there, and whether they're causing bigger leaks elsewhere.
I've actually talked to a few of these guys about sports. I've been in Pres. suites where the Pres. (often from a foreign country who can barely understand the sport he's watching, and who thinks Americans are loony over college sports anyway) is simply deferring to the trustees and alumni. He makes $1m in salary. Don't rock the boat. Regarding sports, he is not making any strategy that benefits the long term health of the university and the welfare of students.
An administrator I know, who is now at the highest levels of a B1G school (no, not Penn State) once proposed a very smart plan for his school's athletic department, but it risked too much for the politicos and the Athletic Director (whose #1 goal was to keep his resume spotless for future promotions to bigger/better schools). I had talked about the administrator's (a Provost) recommendations at dinner weeks prior to that, and boy I tell you he will never get involved with sports ever again. His plan was received, and then people did an end around that chopped off his balls.
This is true especially down south where the coaches and ADs will run to the trustees, pols and boards behind the Presidents back. Down there, you will be fired for anything that can be perceived as inconveniencing sports. Remember Elsa Benitez at Texas A&M. Fired when the AD reported a surprise enormous deficit which she wasn't willing to totally subsidize.
So don't tell me Presidents make a considered analysis to lose money in athletics. They don't. The only thing they consider is how to keep making $1m in salary without permanently damaging the university, and when you hear about a President pulling a Benitez, make no mistake, something is going very, very, very wrong.