<p>There is an assumption underlying this question posed by the OP that there should or can be a single national university and that this university should be the best that the nation can offer.
As many have pointed, each state has its own public university. In many states, this public university (system) is the best or at least one of the best (eg. Cal & UCLA and Stanford & Caltech). But that is not true in every state (Yale vs. University of Connecticut, for instance or Harvard & MIT vs. UMass-Amherst).
Another point to consider: In 2007, the University of Massachusetts system (Amherst and four smaller campuses) had a total endowment of $238 million vs. approximately $30billion for Harvard and around $8billion for MIT. Chicago’s endowment is $6.5 billion vs. $2.2 billion for the much larger University of Illinois system. the cost of creating a first-class national university would be prohibitive, not to mention that the whole concept is unnecessary.</p>