The schools have more control over the Greek system if they welcome them on campus as student groups. Many schools assign dorm wings to certain houses, and thus the dorm rules of no drugs and alcohol apply.
U of Colorado fraternities are not recognized as student organizations. Rush is done off campus. The school imposed a lot of conditions for the fraternities to return as a student group and the frats said no, they’d prefer to operate outside the school’s requirement. It’s worked just fine for 15+ years and the school has absolutely no control over them.
Harvard doesn’t allow single sex organization. There are sororities (and I believe fraternities) off campus, but as of this year no student who joins will receive Harvard’s help with scholarships and grant applications, applications for professional schools, cannot be captains of sports teams or in leadership of any Harvard student club. I think it is ridiculous that an adult Harvard felt worthy of admission cannot be trusted to make a decision to join an outside organization. These same students can join the KKK or NRA or now that the Boy Scouts are co-ed, the BSA, but not a sorority which may give them scholarship opportunities and leadership roles and civic opportunities. We wouldn’t want women exposed to those bad influences, or even give them the opportunity to decide for themselves who they want to associate with.
Greek life at MIT, Yale, Tufts, Northwestern is growing. I guess women at those schools are better at self regulating.