I think I can see where you are coming from on this, the main reason I disagree is that I do not think there is a good line between public and private enterprises. Whatever you do always ends up being a little bit of both, no matter which box it claims to be in.
Also, the state is in the business of managing externalities, especially those for which market information is lacking.
Public subsidies of private businesses can sometimes make a lot of sense from this perspective as they are in effect a form of investment by the actor that is able to take the biggest risks, which given the market runs on historic data, is very useful when someone wants to try something that is particularly novel.
Now it can be a bad investment, either by not being useful, or by driving out other useful things, but this is true of all investment decisions, whether or not it is the state making the investment.
Another thing is that other than possibly in communist states, I am not sure that any country has cut private business entirely out of public projects, and they only did it by banning private business entirely, so I was wondering what evidence there is that your suggestion is a good idea economically.