Explanation of parallel axis theorem

[Trollfaz]

For a rotating system with mass m this theorem says that if it rotates about an axis distance x from but parallel to the axis through it's natural mass center (CM), then I moment of inertia is
$$I=I_{CM}+mx^2$$
My thinking is if one move the axis x distance away from the axis through it's CM, and we can treat the object as a point mass at it's CM, then it's as though we are moving that point x distance away from the axis of rotation, contributing another $mx^2$ moment of inertia, is this explanation correct?

 

[Trollfaz]

So for instance I of sphere mass m is $\frac{2}{5}mr^2$ for radius=r. But in Newtonian mechanics, we can treat the sphere as a point mass in its geometrical center. Then if this axis of rotation is x away from it's CM, then the point mass is also x from the axis of rotation add another $mx^2$ to I. Assuming sphere is uniformly distributed in mass

 

[Orodruin]

But in Newtonian mechanics, we can treat the sphere as a point mass in its geometrical center.

We most certainly cannot. Only for certain things such as the gravitational field outside the sphere does this hold.

In particular, the sphere has a moment of inertia around its CM - which the point particle does not.