So I've learned that the "On Surface" snap just works in the 3D screen. (?)

I was thinking the center of the circle would be submerged, inside the sphere, and the circle would be on the surface.
(Which construct-curve-project already does.)
A script, if possible, would be able to find that center point. Do not know if it is worth the effort though.
I see that UI PropertiesPanel has a read-only radius...