True, with a but.A CPL holder can fly for compensation in a plane provided and controlled by a third party. He is just he buss driver. I have made some nice money flying people around in their aircraft. The only control I over the flight I would exercise is safety of flight items, gross weight limitations, and places where we could legally go (usually international).
The "but" (which is probably the No 1 gotcha) is that the flight itself must be an authorized operation. Many of them are so no problem. But, for example, if that "third party controlled" flight happens to be a charter or improper flight for compensation, the pilot can be dinged for a violation if the pilot knew or should have known the nature of the flight. That's a matter of the available evidence and, hopefully, most cases don't et far enough for the pilot to have to win that battle at trial. But the point is, a third party was in operational control does not automatically insulate the pilot.