Justification for the 2D cut-out method for determining a rocket's center of pressure
I couldn't justify this method more simply than NASA does:


  For model rockets, the magnitude of the pressure variation is quite small. If we assume that the pressure is nearly constant, finding the average location of the pressure times the area distribution reduces to finding just the average location of the projected area distribution.


So the pressure, constant everywhere, "cancels out" leaving you with only area.

And the simplification from a 3-D shape to its 2-D projected area is justified by the shape's relative simplicity and symmetry.  This wouldn't work with a grand piano.