I know more about the RB situation- the Admin said they'd fund it because it would be primarily students and faculty who use it and the Swim/Dive teams would rent pool space. That is why the whole pool fiasco ensued-- when it came time to pay up, Admin went back on what they said, and the AD couldn't help because they weren't authorized to seek donations...

I do know the MC, SFH are also that way- they are buildings that rent out time to the teams. I am unsure of LES, and the IPF (assuming that was ALL private donors). BUT in my experience it plays out like this:

AD: "We really need this building, we are willing to fund raise in order to keep costs from church members, etc."
Admin: "How will this better the spiritual well-being of the students?"
AD: "If the school helps fund a portion we would love to have the students use the facilities. We would also appreciate the use of Church funds to help pay for maintenance (janitorial NOT upgrades)."
Admin: "Only if we dictate (maybe the wrong word choice) who can use the facility at what time."
AD: "Well, we feel as though our Student-Athletes should get first priority."
Admin: "It is a blessing for them to be at BYU, we are gracious enough to allow them to even have a facility. Anymore talk of how the Athletes should be given priority and we are done with XXXX sport."
AD: "Ok, we will do it your way."

Maybe not word for word, but that is basically what happens-- many sports have almost been given the ax because of building issues