To be clear at the top, I'm not challenging the notion that aerospace is a heavily regulated industry/enterprise/whatever. I'm only challenging the idea that regulation is proof of de facto state control. Every business is regulated. I can't open a corner bakery without the government oversight and regulation, but that doesn't mean that my bakery is run by the government.
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.
Do you have evidence that this is true? I've found an abundance of technical specifications on both the design and construction of the F-1 rocket engine, for example. I also cannot find any examples of a law or regulation that prohibits the design or construction of rocket engines (notwithstanding regulations on the components or materials used, like hazardous materials and such), but I'd happily consider any sources you provide.
What evidence would you consider valid proof to the contrary? Technical specifications? Personal testimony? Something else?
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.
Elon Musk claims that he started the company of his own volition after selling PayPal for oodles of billions of dollars. That you personally think investing <10% of that wealth in an unproven aerospace company is too risky is hardly evidence of anything. You're not a self-made billionaire entrepreneur.
Beyond that, nothing you've said is evidence that SpaceX isn't a private firm or that engineers employed by SpaceX didn't design and create its own rocket engines and launch vehicles. I will happily try and provide you with evidence that they did, but first I want to know what sort of evidence you think would be legitimate and sufficient to establish (or at least indicate) the truth of the matter. Personal testimony? Technical documents? Demonstrations of novel technologies and vehicles? Journalistic inquiry? Something else?