tl:dr: it seems difficult to both SNAP >>and<< Align the 3dC to a face center in one action. You can free-hand it very close, but “Snap To Selected” doesn’t align, and Cursor (Tool) will only be as accurate as the mouse hand. If the “Snap To Selected” command can be made to align, I’ll be happy to hear it.
On to my ignorance:
Yeah, I thought the Orientation dropdown ( none/View/Transform/Geometry ) could set the cursor to align to the orientation of the mesh element selected, but I have trouble getting it to work (although I occasionally, like a blind squirrel, manage ) becaue there’s the 3dC orientation, PLUS the creation orientaion of primitives in the F9 dialog ( 3d Cursor/World/View).
I think I recall my suggestion earler: SNAP CURSOR TO SELECTED >>only<< moves the spatial location of the 3dc. But the “Cursor TOOL” does both move the cursor AND aligns it to the mesh element it is snapping too, but is only as accurate as the operator’s mouse hand.
EG: make an object in which the faces have obvious differences, and “Cursor Tool” (we need a verb for that) to one face, add a cylinder as a pointer, then SNAP the 3dc to another face and add another cylinder. If I’m getting this right, the two cylinders will be parallel.
(My RCS proposal was to have the Cursor Tool behavior define the mouse-placed-3dc behavior, for which there is Blender UI precedent in the Sculpt Smooth Tool strength. Since IMO unaligned 3dC is not particularly useful, having the Snap behavior changed would be both more useful and intuitive.)
If SNAPPING to FACE CENTER worked for the Cursor Tool, that would be good, if it were aligned.
NOTE my “in one action”. If you use the Cursor Tool to get the normal and thereby set the alignment of the 3dc, it will “stick” when you use “Cursor To Selected”.