Robert Goddard developed liquid fueled rocket engines with private capital. And all but two of the great telescopes of the 19th and early 20th centuries were built with private money. SpaceX, of course, was seeded with private money. Though I think we'd agree that Elon Musk couldn't have raised enough capital to get SpaceX to where it is now without NASA's help, we might disagree as to why such capital is so hard to come by.
I also agree with you that NASA (the agency) effectively lacks the will to explore. And I would argue, as Robert Zubrin does, that it's because they prioritize safety above mission success. Safety is critically important. But if safety is your primary goal, it's always safer not to fly the mission.
But back to why I think private companies can't raise enough capital for risky endeavors. Feel free to label me a Paul-bot, but in my opinion the root problem is that the federal reserve fixes the price of money, depreciates the currency, and allows a small group of banker cronies to skim the difference between the market rate for money and the artificial fed rate. It's hardly a market economy if the market for money is rigged by the government. The banks have now siphoned a majority share of our economy's free capital. And since they didn't earn it anyway, they gamble with it. But if you look historically at the U.S. when private citizens controlled most of the capital, they did in fact make risky investments and achieve great things. I don't think capitalism is our problem, I think it's the solution to our corporate welfare-ism. But we can't have effective capitalism until we have sound money.
This video contains some interesting history about commercial space development: