Should universities charge rent to 'profitable' athletics programs?

Is it reasonable to revisit the relationship with athletics with more of a hardball, Art-of-the-Deal type approach?
We all understand the basic surface level explanations as to why they "don't owe us anything" (athletics is self-funding, donors gave to expressly to athletics, we should be thankful our school is on the map, etc) but this misses a major economic fact that their relationship to us is that we are the host, and they are the ones who live off the host. There would be no University of Oregon Football Team without the University of Oregon. Thus there is an economic (not accounting!) argument that they do, in fact, "owe us something." We should have some leverage here.
Clearly, we're in a situation in which we (the hosts) are in such dire straights as to require the academic equivalent of selling kidneys (ending tenure), all while the football team is paying out in the $10s of millions in Name, Image and Likeness contracts to our students.
Scholz is an economist, he should understand basic leverage principles. I'm not exactly suggesting a Trump-style shakedown, that we should go say "nice football team you got there, very self-funded, lot of lucrative contracts, would be a shame..." All I'm saying is that, it's nice that we're not saying that, would be a shame if the Overton window shifted and we started saying that.
FWIW, I do totally get why a wealthy donor would be hesitant to throw money into a swirling sea of administrative bloat; telling donors that 4% or whatever they earmarked of their donation goes into the general fund might annoy some donors, but two things: 1) Big 10 schools with storied football teams that wealthy donors can rub their scent all over are a finite commodity, they don't grow on trees. If you want to fund University of Austin football program, go ahead and good luck with that. 2) If you have hate administrative bloat, guess what, University of Oregon is a public institution, and if you're wealthy enough, you probably have more resources than an overburdened and under-resourced faculty committee and might be able to swing a little harder there. Thank you for taking interest in administrative bloat at University of Oregon. Go for it!
FWIW math departments also have the problem of not wanting ear-marked donations getting tossed into the general fund; occasionally graduates with math degrees become insanely wealthy and want to give back, and they don't like being told they have to route this through the general fund.
But at the end of the day there's a basic Render-unto-Caeser principle (at least that's how I interpreted Jesus asking "well, whose face is on the coin?"). If University of Oregon is all over your branding, kick some back to University of Oregon.