SpaceX, naturally.
But being run better than NASA is not exactly a high bar. I would actually amend it to most US federal agencies, even. It's just that Americans, for some reason, assume that the way their government works is exemplar of governments in general. That is not true, to put it mildly - US is just particularly dysfunctional in that regard.
I should also note that my original point was not about a public/private dichotomy, but specifically about the "capitalist hero" cult of personality and the associated management style. You can have private companies that aren't run in this manner, where decisions are collegial or otherwise checked. You can even have private co-ops. And, of course, in many - indeed, most - cases you don't actually need megacorps to do things that need to be done, and a bunch of much smaller competing entities will do just fine without all the undesired political and economical effects.