Please let us not equate those operating outside CAS with "Private Pilots".
Many indeed are, but many are doing one kind of aerial work or public transport or another, and are therefore professional pilots.
Not that it should matter to those giving services, but a mindset that dismisses all aircraft outside controlled airspace as recreational bimblers leaves aside ambulances, pipe and wire inspections, air charter, at some times of year mail and so on.
Which, of course, raises the thorny issues of registration vs flight number and whether it makes a difference.