Of course the Patent and Trademark office is law supported too.
(And, naively, aren't there many private businesses that couldn't make a profit without laws supporting them?)
In a sense, no, by definition.
Would those factors actually make them "not really profitable" or "not actually private businesses"? Or is this where the "In a sense" part of your reply breaks down? (These are legitimate questions, but I'm afraid it might sound snarky.)
Of course, by that standard, I'm not sure how many "truly private" companies exist, since it seems to rule out (among others) any company that relies on intellectual property law, including trademark enforcement to prevent bootlegs and cheap imitations.