Certainly this would be a way for the Holy Father to confront the liberals without getting into politics. I doubt that many of these schools could survive as strictly secular institutions. Most do not rank very high on the USNEWS listing.
Yes, and there is a lesson here for conservatives. Rather than the almost hopeless task of trying to reform the most corrupted of the liberal institutions, we should be thinking of ways to destroy them. Hitting them in the pocketbook by means of constant public denunciation and by means of lawsuits is one way to start. The Catholic church can withdraw their phony pseudo-Catholic seal of approval, and in the secular realm, political activists and lawyers (and angry parents) can do the rest. It would be wonderful to see a few dozen colleges and universities driven to utter extinction in this way.