I had some trouble getting this to be useful, so I constructed a simple test scene to limit the parameters. There might be a combination I’m not trying, what with all the permutations, including coordinate systems. Here it is: Tangent Align.blend (656.0 KB)
So, a simple ‘sphere’, plus two test objects.
I was unable to get the test objects, the small cone and cylinder, to align to Edges in any useful way, although that might be because the normals for Edges are defined unintuitively to me (iow, I expect the Edge normal to bisect the convex angle of the two comprising Faces).
Faces were much better, but even then often the Cone would adhere at the tip (inside the sphere), even though the base was selected as the Active element and “Active” was selected in Snapping Options. This limited the locations at which I could place the cone. This was most noticeable in the 2nd row Faces of the ‘sphere’.
Anyway, placement by hand would be much less time consuming than unraveling the proper procedure to make this work. Onward.