Your understanding is invomplete. They are a completely private company. They get tax funded money for launches and won a tax funded contract to develop american manned spaced fight again, but they still get the bulk from private launches, japanese billionaire manned spaceflight projects, and so on.

The government uses SpaceX like Uber. Just because you ride ubed doesn't make you it's owner. The contracts were big funding but companies besides the government buy most launches.