Because a private UI scheme would probably be even
more expensive than it already is - just like your health care system.
Would private insurance be more "efficient" than government? Efficient at raking in profits, yes. Efficient at taking care of the clients? Debatable.
I'm not trying to stir an argument - honest! All I'm saying is that we can never
assume that any particular activity will be run more efficiently (or provide better service) by private industry, than by government.