The above poster is correct. Just like Lockheed Martin and Northrop Grumman, the space projects are being built by NASA under a different name.
It's a congressional mandate that federal agencies use contractors for most roles and proejcts. Private contractors are seen as superior and more cost efficient than government sponsored engineering. The FBI, DOJ, FDA, CIA, NSA, and all other three and four letter agencies use contractors en mass. There are significantly more people who work for the government through contractors than there are government employees. At places like NASA and NOAA, the only people working directly for NASA are the managers and security.
But this is not to say that the government has no control over its contractors. Government contractors are basically temp agencies. The contracted employees work on site at the government base, under the direction of the government civil servant, answerable directly to the government. They have secret government clearances and take polygraph tests. The only interaction the typical engineer has with his parent company is receiving his paycheck.
Do you have evidence that this is true of SpaceX?
I think that there is an abundance of evidence to support that SpaceX is radically different than you describe; that it is a genuine, private aerospace company originating from a single wealthy businessman; and, that creates and builds rockets. If I presented you with such evidence, that SpaceX is a private firm that employs actual engineers to design and build actual rockets, would you consider it and take it seriously?
If so, what kind of evidence would you consider legitimate? What kind of evidence do you think would be suitable to support the position I described?
SpaceX can't be truly private since rockets which can reach orbit are a classified technology. The government doesn't let that stuff into the public domain. They don't let private companies go willy nilly, building classified technology in unsecured and uncontrolled environments without direct civil servant oversight.
SpaceX company was specifically created to cater to NASA. The impracticality of a truly private space program without governmental oversight is three fold. Not only is it impossible to build orbital rockets legally, it's also impossible to breach military airspace without prior clearance and scrutiny. It's also impractical to invest hundreds of millions of dollars into building a rocket when you don't even know if NASA is going to buy your services or not; whether they would continue using their own rockets, or outsource their space program to another country with launch capability, such as the ESA. Clearly, the deal was struck with NASA before the fact.
Lastly, SpaceX has its offices on government land and the launches are conducted from military bases, which is an overt indication of its status.