MCs do animate but nested animations inside a MC are only seen from the parent timeline after published/compiled. MCs are dynamic and are disconnected from all parent timelines.
GRAPHIC symbols are synced to all parent timelines and render inside of Animate - most animators use graphic behavior for this very reason and because we can control them precisely via the Properties Panel and Frame Picker Window.
Now for blurring - you can apply a filter to a MC, yes, but you can also apply a filter to a layer. So you can move the symbol to its own layer where you want the blur to take place, and add the blur effect to the layer during the motion you are trying to create.