I can definitely say from personal experience with direct knowledge of some department operations that is not always true. For instance, I can tell you that straight from the source(s), there are basketball programs with multi-purpose arenas where the department charges a set fee for time share and deducts it as an expense against that program for accounting purposes. There are several that I'm aware of that do this. Don't know the percentage, but it's fairly common in my experiences. I definitely think you're overstating the academic side paying for the facilities; I can't count how many times I've seen facility renovations/projects based solely on donations, fund-raising, etc. The academic side sometimes isn't involved in facility projects.
The facilities don't lead to anyone profiting, though -- that's something I agree with. But to the extent the academic side is financing those facilities, they're doing so because it's a good loss leader for exposure. One study I read that was performed a couple decades ago showed that at state universities, athletics led to another 1,000 students attending on average, and success of football and basketball specifically increased that even more. Universities do finance some of the athletic facilities (though like I said, I know of several cases where the athletic departments are paying the expenses on the debt), but they're doing it because it's helping them. It's just not traditional profit.
I don't necessarily have a problem with coaches making that money, so I don't want to say they make "too much," but if it comes down to finding a way to pay a stipend, I'm merely saying it could be done. When I studied this a few years ago, I found coaches' salaries comprise about 30 percent of expenses nationally among all Division I schools. The average assistant coach in FBS for all sports was bringing in roughly $150,000 (though obviously non-revenue sports would have been less) with the average head coach averaging over seven figures.
I'm definitely not against paying coaches a cozy salary. All I mean to say is that there is a lot of additional money being reinvested in salaries and facilities that could be spent on miscellaneous aid if they really wanted to do so.