Ask your alma mater's department heads, deans and development officers if sports matters. Research (even tech transfer) doesn't raise enough cash to fund our current university system. Tuition doesn't either; and state support (if applicable) doesn't get far either.
Donors provide the rest. The best way to butter up a donor is through university brand prominence. A non Ivy League school can't compete in the same fields as an MIT, Stanford, Cal Tech, etc. You have to increase your university brand through any respectable means necessary. Sports is often that angle.
Go ask your endowed profs if sports matters in any factor to their compensation. It's likely they'll admit that it helps, at minimum.