The Supreme Court has also held that the government can revoke tax exempt status of a private organization where it furthers a compelling government policy: https://en.wikipedia.org/wiki/Bob_Jones_University_v._United...
Control over federal funding is also the hook for Title VI’s application of non-discrimination laws to private universities.
The government also has the trump card up its sleeve that Harvard is almost certainly violating Title VI through extensive programs of race consciousness. It’s well established that the civil rights laws apply equally to whites as to non-whites. Harvard has many programs for non-whites where, if those programs were for whites instead, that would be a Title VI violation that would jeopardize Harvard’s federal funding. E.g. Harvard had various racially segregated graduation parties last year: https://www.nationalreview.com/news/harvard-university-to-of.... If you can’t have a “White Celebration” then you can’t have a “Black Celebration” either. If Harvard doesn’t settle they’ll get hit with a Title VI lawsuit and they’re going to lose it.