EDIT: More briefly than before . Here is a naive physical argument which might meet the  request of *a point of view from which the result would immediately appear to be just what one would expect before going through the argument.*

 - A blob of (incompressible) fluid volume V will form a spherical ball of radius (what it needs to be) if uncontrained
 - A blob of fluid volume V constrained between two parallel plates at z=r and z=-r will form the height 2r central slice  of a sphere of radius R where R is just right so that the slice has volume V (provided r is not too large relative to V in which case we get a sphere)
 -  A blob of fluid of volume V constrained between two parallel plates at z=r and z=-r and with a cylinder of height 2r and radius q imposed in the middle will form (along with the cylinder) the height 2r central slice  of a sphere of radius R where R is just right so that the slice has volume $V+\pi q^2 h$. This shape might not have the curved boundry reach the cylinder
 - Imagine that the volume is just right to get that napkin ring. Now start shrinking q. We will still have a napkin ring. Keep going until q=0 and we see that the volume was that of a sphere of radius r.

 
(read the previous version if you wish, it might not be worth it)