Delta Executives typically fly on Delta just as any other passenger (or employee on a positive space pass). Richard Anderson flies in coach and makes no special arrangements to accommodate him. If he were not recognizable and approachable, one would just assume he's a regular Joe business traveler.

Delta Private Jets was part of Comair going all the way back to their 91/135 roots. Today it is largely performed by contracted vendors and used to cross market with Delta's premium travel offerings.

Prior to this executive team, Mullin and crew did use off line charters. I don't know if they used Delta Private Jets, but they sure could have.