Prominent University-educated rabbis of the past (many of whom can be called gedolim):
Rambam (Islamic philosophy; University of al-Karaouine, Fez, Morocco)
Sforno (Medicine; University of Rome)
Rabbi Dr. Joseph Solomon Delmedigo (Medicine; University of Padua, which had granted degrees to hundreds of observant Jews before the 19th century)
Rabbi Samson Raphael Hirsch (Undergraduate study, University of Bonn)
Rabbi Dr. Ezriel Hildesheimer (Undergraduate study, University of Berlin, and PhD in Bible from University of Halle-Wittenberg)
Rabbi Dr. Bernard Revel (PhD from Dropsie College; dissertation on the Karaites)
Rabbi Dr. Yitzhak HaLevi Herzog (PhD from University of London; in his dissertation he identified the source for techelit)
Rabbi Dr. Joseph Soloveitchik (PhD in Jewish philosophy from University of Berlin)
Rabbi Yitzchok Hutner (studied Jewish philosophy at the University of Berlin; Rav Soloveitchik was a classmate!)
Rabbi Avigdor Miller (Undergraduate degree from Yeshiva College; it would not be fair to Rav Miller’s memory that later in life to fail to mention that he became an opponent of that institution)
And the list of living rabbis with university education is very long.
I of course am not fit to argue anything with Rav Shach z’tz’l! But others of similar stature did not agree with him.