AOTW, your statement that the aircraft can pivot about any point other than the CG is a bit questionable.
The "turn about the nose" and "turn about the tail" manoeuvres are co-ordination exercises where the whole aircraft is moved, using all controls, to make the nose or tail stay in one spot. Might as well fly it around a square and then say that the cg was in the centre of the square.
You gotta look at just the forces on the acft. And they do all tend to act about one central point, which can be the CG, or (like aerofoils) the Centre of Pressure, or the rotor head.
Look at the videos of a helicopter gyrating around with loss of T/R thrust. Some forces acting around the CG, others about the rotor head, and when those 2 don't line up, you get roll coupling and it spins towards its side and rolls up big time.
Toss a stick into the air. Apart from the translation caused by the toss, does the stick spin about its end? No, it spins about the cg. Tossing a caber looks like a spin about the end, but it is a combination of a turn about the cg and the movement of the cg in the air.