All countries have a mix of public and private - the question is what is the optimum proportion, and how do you optimise the contribution of both to the public good.  As I understand it Keynes didn't factor in the cost of non-renewable externalities, neither where individual polities played off against each other by mobile capital to the degree to which that happens today.  I'm no economist, but I'm also not sure what contribution Keynes had to make to the optimisation of public sector efficiencies in meeting agreed objectives.  However the universal reference point of "the market" provides some somewhat spurious measure of that for the private sector as far as the neo-libs are concerned, and that is part of their effectiveness in public discourse.

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