Lonewolf
I can't think of an airline in operation that would not have an admin frequency at the operating base that allows a flight crew to communicate via radio to the company (be it for admin, ops, scheduling, or maintenance issues) without having to go through the tower. There are typically multpile radios available to the flight crew to use.
For at least since 1996 with the arrival of the B777, all this type of communication is done by satfone. If you have not used it, it is exactly as clear as any landbased phone. The days of calling base by radio or HF are long gone from long haul airplanes.
I endorse your clear explanation of prioritising and the timeless validity of the dictum "aviate navigate communicate"
Coireall
Sorry for repeating what was said in #1330 and #1331