As for the elite=exclusive argument:
Eh.
UMich has graduated more students than some small countries and states have total people (over 600K and counting), so not exactly exclusive.
Yet, while not an Ivy (though @Alexandre would tell you that the student experience there is similar to the student experience at Cornell, which is an Ivy), it is seen as elite in many fields.
Waseda in Japan has graduated even more students. It is seen as one of the top unis in Japan and one of the top 2 privates (along with Keio; the Waseda-Keio rivalry is akin to the Harvard-Yale rivalry here).