It is somewhat depressing that when you accidentally make a comment that attracts a lot of downvotes, you can't even delete it and have to watch them rolling.
The truth is that by a literal definition of the words private equity, SpaceX does qualify. All private companies are equity that is private. But when people use the _phrase_ private equity, they're referring to people who are specifically starting with a pot of money and then shopping around for companies they think they can flip for a profit, often with the normal trappings of private equity, for example, `carry` and particular tax and business strategies (debt, etc). So because SpaceX was started by a founder with an actual mission and grown from zero, and because their goal is primarily to build rockets and not make a quick buck, most would simply call it a 'private company' to avoid confusion.