Intersting topic this...I noticed in the opening gambit to the topic that CFS were mentioned. For what it's worth, in my experience both as a stude on several piston/jet aircraft and as an instructor on Tucanos I can quite honestly say that in a single engined aircraft I have never shut an engine down for demonstration purposes or had one shut down on me. (I cannot begin to imagine the subsequent board of enquiry if anything went pear shaped).
The only occasions where engine shutdowns are carried out (in my world anyway - 2 engined FJ) are on convex...one trip only to give you confidence in the ac flying on 1 donk (med/hi level to allow for several relights should the first not succeed) and on engine airtests.
Shutting down your only serviceable engine in flight is, I would say, pretty poor airmanship and could result in egg on your face.