The University of Chicago plays football but at the DIII level. Here is a list of universities that are among the Top 30 in endowments who do not play D1 sports:
MIT (#5)
Emory (#9)
Washington (St. Louis)(#12)
Chicago (#13)
Johns Hopkins (#16) - they play D1 in a couple of sports but not football or basketball
NYU (#21)
Cal Tech (#23)
Williams (#24)
Amherst (#26)
Pomona (#28)
Rochester (#29)
Most of the others in the top 30 do not play FBS football.
Obviously, you can get people to donate to your institution without playing high level sports. I suspect that most of those who donate to universities because of their sports programs donate most of their money to the athletic programs.
Let me underscore my point. Treating your team sports (football, basketball, volleyball, hockey, etc) poorly upsets more people than individual sports because there are more people to upset. The flipside is that when your team sports do well, you have more people who want to donate.
Academia doesn't understand why people donate. They mistakenly believe it is only about academic excellence in their field. Most of their fields have a smaller visibility than sports teams. Basic name recognition is crucial to donation. UMN has done a wonderful job getting their name recognized... for doing unethical, morally questionable, and even criminal behaviors... both academically and athletically. It's no wonder people don't want to come to UMN for anything because all they hear is negative, in big and small stories. They try to push positive academic stories, but because they are so small, realtively speaking, most people forget about them because they don't impact their lives in the short term.
For good or bad, sports, especially team sports, foster greater inclusion and sense of community because they have immediately visible results. Academic results coming from academic excellence are not immediate results. They might be longer lasting, but don't move masses of people to support a school the way sports does.
The idealism of Academia doesn't put a good feeling in the hearts of people on a bad day at work. It doesn't make a kid dream that there might bigger opportunities available to them. Those are the things non-school connected people relate to.
People don't donate en masse to a school because that school wrote new code for a computer operating system or created a new method for cardiovascular surgery. They need a more visible reason, sports is that reason.