I checked the neutral safety switch and it works correctly. I'm glad because I would not look forward to sourcing and replacing that.
After a trip to Sears to purchase a 5/8" obstruction wrench (I've never had need of one before, but there's no way the top fastener can be removed without one), I was able to remove the starter and test it. (The image is rotated 90 degrees counter-clockwise.)
The solenoid will engage as it is supposed to do, but the armature will not spin. That piece of tape sticking out was wrapped around a brush lead to keep it from shorting out on the case. As far as I can tell, while the tape was loose, it was not shorting out.
So I'm soaking all the threads in Kroil (I love that stuff) and looking for a re-build kit.