Hi Matt @Marxistfenix ,
In Object Mode, the Bounding Box of the Sphere is used to Snap with, so Snap Base set to Closest, gives something like this:

What you can do (since Blender 4.0), is to set the Base with which to Snap manually with B (this will override what is selected in the Snap Base Settings).
To do this (and it might take some practice), G to Move the Sphere, then B, to set the Snap Base (choose a Vertex on the Sphere):

Then Left-Click (or ENTER) to confirm and Snap by moving your Mouse Pointer to the Vertex you want to Snap to:
Like I said, it takes some practice, but it is super handy and takes Blender’s Snapping to a whole new level.