It is simply not feasible to produce public goods using private funding. The reason for this is that there is no way to prevent free riders.
A public good is non-excludable. That means that there is no way to prevent someone from using it. Private goods are excludable. Someone cannot use it unless they pay for it. With public goods, this is impossible. For example, clean air is a public good. There is no way to prevent one person from using the clean air because they did not contribute to some private clean air fund. As another example, you cannot provide national defense to one person but not to their neighbor who did not pay his or her fair share to a private national defense fund.
Private funding will not produce public goods because there is no way to force people to pay for the goods in order to use them.