There's more to this question than first meets the eye at first-- it opens to the door to a question about what do we mean that we say an object is "rotating about its CG".
Do we mean that if we have a movie camera following (or travelling above) an aircraft in linear flight, such that the CG of the aircraft is in the middle of each movie frame, and then "something happens"-- a gust of wind strikes the airplane, or the pilot makes a control input-- the CG of the aircraft will stay in the middle each new movie frame as the camera continues travelling along linearly?
If that's what we mean, the answer is "no"-- that wouldn't happen when a gust of wind struck an airplane, or when a gust of wind struck a weathervane that was somehow levitating in the air. It also wouldn't happen when the pilot pulled back on the control stick or stepped on one rudder pedal.
In many maneuvers we can identify some other point, often far beyond the physical edges of the aircraft, as the center of rotation of the maneuver. Obvious examples include perfectly round turns, or perfectly circular loops. If we are only interested in describing the actual motion of the body, and not the forces and torques involved, then we can arguably describe these maneuvers as simple rotations around the point in the center of the maneuver, with zero translation involved.
Similarly, this related answer considers the simple case of describing the motion of a pencil being lifted from a table while keeping one end on the table-- we can say the pencil is rotating around its end with zero translation, or we can say the pencil is rotating around it CG while also translating diagonally.
However all these alternative descriptions that take some point other than the CG as the center of rotation fail to accurately describe the linear acceleration of the body in terms of the net force acting on the body.
The Center of Gravity is unique in the sense that it is the only point on the body whose instantaneous linear acceleration is directly proportional to the actual net force acting on the body, while the instantaneous linear acceleration of all other points on the body can be viewed as the sum of the instantaneous linear acceleration of the CG plus a rotational acceleration about the CG.
If we define rotational acceleration as occurring around some point other than the CG of a body, then we'll find we have to introduce a pseudoforce representing the gain or loss of "apparent weight" due to G-loading, acting at the CG, which is equal in magnitude and opposite in direction to the real net force acting on the body. Now the forces we're considering to be acting on the body (including this pseudoforce) no longer add up to be equal to the actual net acceleration of the body (times mass).
It's not uncommon in aviation to find diagrams that take exactly this approach. See footnote 1 for a specific example. Such diagrams are not necessarily "wrong", as long as their limits are understood-- they do not correctly depict the actual net force acting on the object, and thus they do not correctly depict the instantaneous linear acceleration of the object.
If the linear acceleration of a body is zero, and the rotational acceleration of that body is also zero, then need to consider a "pseudoforce" acting at the CG vanishes, and any point will serve equally well as a "pivot point" for torque calculations.
One way to think about the "pendulum effect" conundrum (see this related ASE question) is to recognize that if we select some point (such as the wing's effective "center of lift") other than the CG as our "pivot point" for torque calculations, then the "apparent weight" vector acting at the CG can contribute a roll torque about this point. The dynamics of turning flight and sideslips are such that this roll torque contribution from the "apparent weight" vector is generally stabilizing (tends to roll the aircraft toward wings level) if the CG is located below our chosen "pivot point". Naturally, the same choice of "pivot point" may also eliminate other roll torque contributions (e.g. aerodynamic sideforce generated by the wing in a sideslip) that would also be stabilizing-- since the choice of "pivot point" for our torque calculations can not ultimately not affect the answer that we get.
"Shifting gears" a bit, if we are simply trying to describe the motion of an object without regard to the forces and torques at play, there is one sense in which there is one particular point other than the CG may sometimes be the most descriptive choice for some purposes. Consider a steady-state turn with constant yaw and pitch rotation rates. A pivot point can be chosen for each axis that indicates that the point where the undisturbed relative wind is (or would be) parallel to the longitudinal axis of the aircraft. This point may lie outside the physical boundaries of the aircraft itself. See for example this section entitled "Long-Tailed Pitch Effect" from the John Denker's "See How it Flies" website. Since the aircraft is rotating in the nose-up, tail-down sense in the pitch axis, this induces a curvature in the undisturbed relative wind that would be sensed at any point on the aircraft. (For an extreme case, think of how yaw strings or telltales would deflect if mounted on the wingtips of an aircraft in a flat spin, or on the tips of a propeller or rotor blade-- all the strings or telltales would not be parallel to each other.) Since the longitudinal axis of the aircraft is straight, not curved-- the fuselage doesn't "bend like a banana" to conform to the curvature in the relative wind induced by the pitch rotation-- the relative wind can't be parallel to the longitudinal axis along the full length of the aircraft. In this particular diagram, the aircraft has been drawn in an orientation where the relative wind is nearly parallel to the aircraft's longitudinal axis at the nose, but not at the tail. In a sense this diagram depicts a situation where the aircraft is rotating "around" a point near the nose, so the resulting aerodynamic damping effect is not greatly affecting the angle-of-attack of the wing, but is creating a strong positive angle-of-attack at the tail. However, if we simply increased the angle-of-attack of the whole aircraft a few degrees, the point where the longitudinal axis is parallel to the curving relative wind would shift forward to some point ahead of the nose, which would arguably be a more realistic depiction of actual flight with a positively lifting wing. So we may find it useful for some purposes to describe aircraft's motion as a pitch rotation around that point ahead of the nose, plus a translation parallel to the direction of the curving relative wind at that point. The same perspective can be applied to yaw rotation in turning flight. In this section entitled "Long-Tail Slip" of the "See How it Flies" website, we are looking down on an aircraft in a turn, and again we can see how the curving relative wind cannot be parallel to the aircraft's longitudinal axis along the entire length of the fuselage. In the upper figure (8-9), the relative wind is depicted as parallel to the fuselage near the tail, and coming from the inside of the turn at the nose. In the lower figure (8-10), the relative wind is depicted as parallel to the fuselage near the nose, and coming from the outside of the turn at the tail. In a sense, by simply adjusting the aircraft's yaw attitude in relation to the relative wind, we've moved the location of the axis "around" which the aircraft is rotating from the tail up to the nose. Along the same lines, consider what happens if we yaw the aircraft 5 or 10 degrees further toward the "outside" of the turn so that the airflow is now striking the "inside" (left) side of the fuselage along its entire length, so that we are have a slipping turn. One way to describe the aircraft's orientation and motion is to view the yaw rotation as occurring around some point well behind the aircraft, while the aircraft is also translating forward parallel to the direction of the curving relative wind at that point. Conversely, in a skidding turn, where the entire "outside" (right) side of the fuselage is exposed to the relative wind, the point where the point where the curving relative wind would actually be parallel to the (extension of) the aircraft's longitudinal axis has moved to some point far in front of the nose of the aircraft. Now the aircraft can be viewed as rotating (in yaw) around that point, while also translating forward parallel to the instantaneous direction of the relative wind at that point. But these are only descriptions of motions, not of the forces at play-- the net centripetal force actually produced by the aircraft at any instant corresponds to the force needed to move the CG through it's arcing path through the sky, not some other point far ahead of the aircraft or far behind the aircraft.
Footnotes:
For example in the book "Basics of R/C Model Aircraft Design: Practical Techniques for Building Better Models: Practical Techniques for Building Better Models" by Andy Lennon, we find a diagram showing the balance of torques in steady-state turning flight. The center of lift of the wing, not the CG, is taken as the "pivot point" for the analysis of rotational acceleration. The diagram correctly shows the net torque on the aircraft to be zero, but only because a force labelled "centrifugal force" is introduced, pulling downward at the CG, in addition to the actual weight vector. This is the G-loading pseudoforce mentioned above. The torque acting on the aircraft can indeed be correctly analyzed this way--the diagram correctly shows that the net torque is zero-- but the net force (if we include the pseudoforce) is now zero, and no longer reflects the actual net force that must be acting on the aircraft in turning flight. This issue would be avoided if we took the CG as our pivot point for our analysis of rotational acceleration. Lennon then incorrectly goes on to state that aircraft stress analysis often suffers from the "error" of failing to consider the "centrifugal force" imposed by maneuvering flight, considering instead "only" the actual lift force created in the turn or loop
This is especially relevant when we are analyzing the pitch and roll torques created by a pilot's body on a hang glider. Both of these two methods are equally valid: 1) Assuming that at any given instant the pilot is holding himself rigidly fixed in some particular desired position, note the resulting CG of the glider-pilot system and analyze the resulting pitch and roll torques created by the aerodynamic forces acting around this pivot point. 2) Take the CG of the glider alone as the pivot point, and in addition to the aerodynamic forces generated by the glider around this pivot point, also consider the pitch and roll torques created by force exerted by the pilot's arm muscles on the control bar, and by the pilot's "apparent weight vector" (not purely vertical, yet also not necessarily perfectly "square" to the wingspan) acting at the point where the flexible "hang strap" connects to the rigid structure of the glider.
