Let's assume we are dealing with gravity. The gravitational field of a uniform sphere with some mass will only depend on the mass and the distance away from the center of the sphere $R$ (assuming $R$ is outside of the sphere). I want to derive this fact by investigating a spherical shell and showing its gravitational field is equivalent to having all the mass concentrated in the center without using non-obvious facts like Gauss' law.
In other words, I want to show that the following integral is invariant wrt. $r$, the radius of the spherical shell; and inversely proportional to $R^2$ if $R>r$ and zero if $R<r$
$$\int^{2\pi}_{0}\int^{\pi}_{0}\frac{R-r\cos\theta}{\left(\sqrt{R-r \cos\theta)^2+(r\sin\theta\cos\phi)^2+(r\sin\theta\sin\phi)^2}\right)^3}\sin\theta\ d\theta\ d\phi$$
Where the point for which we calculate the field is assumed to be on the $+z$ axis, $\theta$ is the angle made with $z$ axis, $\phi$ is the azimuthal angle on the $xy$ plane.
Seems like a very difficult integral to solve naively. I would love if there are intuitive explanations for this fact, maybe using symmetry etc.