With versions 13, and 14 SurfaceIntegrate simplifies a lot of writing. Is it possible to use spherical or other non-Cartesian coordinate charts. Here's just a simple example:
SurfaceIntegrate[
Grad[1/r, {r, \[Theta], \[Phi]},
"Spherical"], {r, \[Theta], \[Phi]} \[Element] Sphere[]]
