Not on a national scale, no, unless you count any passenger rail or air systems as "successful" and "private." But that's not really a sufficient argument. Obviously, it would be nearly impossible to compete with taxation and eminent domain at the scale they are used in modern developed nations.
> Showing that public roads lead to inefficiency and perverse incentives doesn't point to a solution. It just points to a problem. You still have to find a solution that works.
What do you mean I have to find a solution that works? Does that mean I don't get to propose new solutions and argue for why I think they should work?
> I think this argument is bit of cop-out, well we can't make argument how libertarianism would work so we can poke at something government subsidized and tell everyone how it is broken.
I can make an argument for how libertarianism (or rather, a specific societal organization that some might consider to be "libertarian") could work.
> To me it seems this is not unlike Communism argument...
Well, I don't think communism works well in theory or in practice. I am advocating for markets, and there is a great deal of theoretical and practical evidence for why markets perform well.