Personally, let's let high schools worry about "amateurism" and Title IX. College athletic organizations should be independent investments made by colleges. Let them determine if the return on investment is worth it.
Stop giving "scholarships" and other educational benefits like room and board to escape from the Title IX restrictions. Provide players with an offer that you are willing to give (enough that would actually cover the cost of going to school), and what you believe they are willing to accept. If they come back with a counter, then YOU as an athletic department have to decide if that is worth it to you. You can even make it contingent on a per game basis. If a player chooses to sit out, then you don't have to pay them for that game. Cap rosters and don't allow unlimited "walk-ons". Find a way to restrict poaching if a player wants to transfer, such as a transfer fee. Schools can give counter offers if they want to keep the particular athlete, but schools are not allow to cut players for lack of talent if they commit to that player.
If non-revenue sports decline because of that, then it was a (obviously) flawed economic model that was supported only by government intervention. If schools or benefactors choose that they want to keep softball or gymnastics just for the sake of putting athletes in the Olympics and getting the university's name out there, then that is also a decision that they will have to decide is "worth it".
An Ivy League university can choose to offer admission to anybody that they feel can yield influence in the world. This is why so many celebrities have degrees from Harvard like Ashley Judd. Yes, she is eloquent, but did she score in the Top 1% of admissions tests, or did she get in because she is an actress? It is up to the schools to determine if their reputation is worth admitting that person.
You could argue that the schools with the most money and resources could just buy the best teams because they have the most money to throw around. This is how it works anyway. My only concern is that the Ivy League, with their huge endowment funds and powerful benefactors, might actually be great in athletics if they choose that it is worth the ROI.