If the U.N. controlled the Internet then the oppressive censorship of China and Iran would pervade the whole internet instead of just the parts of it that were in their countries.
As long as the Internet is controlled by the U.S. its control will be farmed out to a private company as it presently is. This private company can be as international as any nonprofit assembly such as the U.N. while not being mired in beuracracy. If they do get mired in beauracracy then the U.S. will not renew their contract and another company will take control.