The answer appears to be a resounding, “No.”
If you look at the American-born CEOs in the Fortune 500 in 2015, only 30 went to elite colleges. The remaining 470 CEOs at the 500 largest companies in the U.S. went to a wide range of non-elite schools.
Of course, elite private colleges and universities have advantages. They have internationally acclaimed professors and cutting-edge curricula and low student-to-faculty ratios so presumably you get a better education and more personal attention. They have loyal alums and talented fellow students who can be a valuable career network. They have a well-staffed Career Services Office, which can guide you on the transition from college to career. And let’s not forget the lovely ivy-covered buildings, libraries and museums. (There’s even an arms race going on at some universities in the U.S over fancy recreation areas with water rides, spas and massage clinics.)
Above all, you are aligned with a prestigious brand whether it is Harvard or Cambridge or France’s grandes ecoles, which will provide a halo to your personal brand particularly at the beginning of your career.