Does aircraft allways rotate about center of gravity like weather vane about pivot point?
It's purely a mathematical trick, but a very good one at that. In reality, pure rotations around the center of gravity are rare.
The key is that it's often convenient to split movement of a body into translations and rotations. So we need some definition to distinguish between a rotation and a translation. For example, consider lifting a pencil from a table on one end. Intuitively, there's definitely some rotation as the pencil is now at an angle, but is there also a translation? The end of the pencil resting on the table has not translated at all, while the end you picked up has translated a lot. So we can describe the same situation by pure rotation around the point resting on the table, or a combination of a translation of the point you're holding and a rotation around that point, or anything in between.
By defining rotations and translations with respect to the center of gravity, you simplify a lot of equations, and you can analyse the system with a free body diagram. If all forces in one direction sum to exactly zero, you will have no translational acceleration, and only rotational acceleration. In reality of course, forces rarely sum to exactly zero, so there's always some translational acceleration of the center of gravity.