real answer: it uses two facts:

- the arbirtrary movement of a rigid can be seen as a translation through any of it's points $P$, whose image is $P'$, and a rotation by some axis passing through $P'$. **Valid to any point $P$ on the body**. (Chasles theorem)

- the center of mass of a rigid body can be seen as a point of the body (with mass zero, only kinematically), meaning it's distance from the rest of points remain unchanged (easily checkable with previous fact and definition of COM).

So here it is: an infinitesimal movement of a rigid body can be understood as a translation and a rotation through (an instantaneous axis that passes by) the center of mass. If the net external force is zero, than the total acceleration of the COM is zero (famous result), but, as the center of mass in on the instanteneous rotation axis, it only has translational acceleration. Thus, the translation acceleration of the system is zero (with respect to the translation of the center of mass at least).