No. If you distributed the students admitted to Harvard among other universities their education would be impaired by the fact that the professors would have to dumb down the material taught in their classes to their less bright colleagues. Not to mention that the professors themselves would not be as smart and accomplished as Harvard professors.

I think you are overestimating how much influence professors have (something I lament as a former professor). The high achieving students succeed despite anything their professors might have done or said.

"the professors would have to dumb down the material taught in their classes to their less bright colleagues."

A fascinating side effect of open courseware and online lecture videos is the elimination of that peculiar historical belief. For example, per video evidence, the kids in Professor Strang's linear algebra classes are definitely not observably smarter than the kids I met in my linear algebra class. There is a small measurable quantitative difference, but in practice there is no observable qualitative difference. The much maligned concept of human biological differences observationally do not extend to mere ivy-attendance.

Counter-anecdote:

I passed University physics by watching MIT OpenCourseWare lectures and those kids learned in a week what took us three...

A distinct flavor of hubris arises from self-proclaimed brilliance being so easily impaired by a silver spoon that isn't quite as shiny.

Yet it is certainly true. More than the professors, I think the peer group matters in pushing students to their full potentials.

It probably has more to do with the simple name of the university opening doors.

Investors and employers will give more to a Harvard grad than to an equally accomplished state-school grad.