The problem wasn't NASA. The problem was, NASA was told to build something impossible - a cheap and reliable spaceplane. It was many times more complex than anything else ever made (including the Saturn V). It was never going to be cheap and reliable. (Well, it might be possible today, with decades more experience and technology). Only the "yes men" said it was possible, and they got promoted. The rest of the engineers were forced to work in a bubble where a realistic big picture didn't matter.
OK, you can privatize that. But the same problems will apply. Look at Australia's military acquisitions. Every vehicle we buy costs 10X the MOTS price, because there's always a requirement "modifications for Australian conditions". The private operators read this as "complete idiots in command", and charge accordingly. A privatized space sector would work the same way - stupid government orders will get stupid prices.