Universities then proceed to all pick being elite.
Why would they do that?
Universities will fill economic niches. There can only be so many elite institutions with high national/international focus, so they can't all do so. Many universities will not even bother to try: they'll fill the other really obvious and important task, which is specialising at providing for the educational needs of the local/regional workforce. But there won't be this dichotomy either, because people who don't get into the elite universities still want a better than average degree, so this will create a market for "mid-tier" universities: essentially ones with superior national / international reputation, research, etc. compared to the local/regional, most of which will never be able to compete with the elite.
And this is what we already have, because the market already makes it happen.