NASA spent many years developing processes and vehicles that, initially, had a very high failure rate. Today we are conditioned to expect that these launches will go off safely, but do we need to give the private sector 20 years or so to sort itself out before we start allowing people to fly in their rockets?
Almost all the US space program is ALREADY private. The dominant US launch platform is United Launch Alliance, which was created by a merger of Lockheed Martin and Boeing when the US tried to bring in competitive bidding for Air Force launches. Their merger allowed them to become the sole bidder on most launches, allowing them to continue to receive their cushy "cost plus" bids. These "cost plus" bids allow them to pretty much charge whatever they want for costs, and then receive on top a guaranteed profit margin. There is no incentive for efficiency. In fact, ULA has almost no international customers because their costs are more than double their competitor's prices. They grow fat at the teet of the US Air Force.
Space X is a relative newcomer, but since its first generation rocket (Falcon 1) had three failures, the last of which was caused by a timing error in stage separation, it has had a nearly perfect launch record. The Falcon 9 rocket has never had a significant failure (ie. one that prevented orbit of the primary payload). Touch wood. Their rockets have been designed in house with the eventual goal of re-use. Their rocket engines are designed to be simple and failure resistant. And their rockets have nine engines, so that even if one fails the flight can continue. I wouldn't write off competition yet.