Could the center of rotation of a spinning aircraft move away from CG?
In some sense, rotation is rarely around the centre of mass. It is extremely convenient to consider any motion as rotation around CM plus translation of that CM, but it doesn't have to be so.

Take the most basic control input for example: from level flight, you pull up for a short climb. A typical explanation of what happens says that the tail produces a pitch-up moment, which rotates the airplane around its CM to a higher angle of attack, this increases lift and it starts to climb. However, a more detailed analysis will point to the fact that initially, the tail will produce an extra negative lift. This is not just a pure moment, this is an unbalanced force applied away from CM. As a result, the airplane will lose some of its lift and will start sinking. Only later will lift grow and pull the airplane up.

Now, if you consider it as a single motion (which it is, fundamentally), you can rightfully say that it is a pure rotation about a point ahead of CM. (It would be nice to draw a diagram here, but not now). The pilot will perceive it as such. On some designs, particularly with delta wings, this initial loss of lift from elevons is so strong that the virtual point of rotation can be ahead of the aircraft (and thus the cockpit). Space shuttle reportedly had such configuration. For pilots, this creates a weird sensation that the aircraft behaves contrary to the expectation, particularly on landing where perception of altitude is acute. This can lead to pilot-induced oscillations.

So, for any motion, not only spin, to say that rotation always happens around CM is a simplification. To be able to say that and remain valid, you have to consider all the translation effects as well. In most cases, this is still the easiest way, but if you want to speak of rotation alone, remember that the instantaneous centre of rotation rarely aligns with CM.