And if you are letting untold numbers of strangers have the run of your "Private" campus, like on graduation day, it is no longer private.
Notre Dame University is a cesspool and is off the rails morally speaking, but that does not that NDU does not enjoy private property rights.
Can you cite anything to support your assertions about the law? You have asserted 2 grounds upon which a private college in the USA has lost its right to eject trespassers, both of which should trouble constitutionalists and people of faith:
1. A private institution that offers any kind of service or program whatsoever to the public is a de facto “public accomodation”, and as such has no right to eject trespassers.
2. A private institution that holds a large ceremony that thousands of people will attend loses its right to eject trepsassers.