Agree in general, though Von Braun himself wasn't in the private sector. All of his work was sponsored by the Nazis directly. With all the nationalistic fanfare surrounding the moonshot, the man directly responsible for putting us there is an SS major, and the engines that got us there are directly descended from V-2.
It's true he pretty much came directly out of university into the Nazi party. He wasn't the only person responsible for NASA's success though. It took a lot of engineering talent from several different fields to make the Apollo program happen.
Including some free market talent from Russia. The physics used to launch the rocket into space in the first place came from my great*3 grandpa Tsiolkovsky, and I've heard (hearsay) that some of the calculations and mechanics for the actual moon landing came from some poor destitute and unrecognized chap in Russia who died before his papers were found and put into use by NASA.