Consider a situation where a wheel is rolling without friction on a level surface. Call the center of the wheel $C$, the point where the wheel contacts the ground $G$, and some arbitrary other point on the rim of the wheel $P$.
The normal force has no torque when $G$ or $C$ is picked as the axis of rotation, but it does if $P$ is chosen for any point (minus the top of the wheel). If a physics problem then says to pick the force that gives the largest torque, how am I meant to choose if the axis of rotation choice changes the answer?
The physics book I have states that $G$ or $C$ are points at which an axis of rotation may be placed because they both have the same angular acceleration around each other. This seems fine to me; I can visualize the concept of the $C$ rotating about $G$. The trouble is that it seems to be the same situation for any $P$.
Working out the math, it does seem that every point on the rim chosen as the axis of rotation is such that the center of mass rotates about the chosen point at the same speed as the chosen point rotates about the center of mass.
What is going on here? How does one 'get a feel' for these problems where it seems that the axis of rotation choice is so critical?