I can't read the posts of the posters that con-pilot quoted, as I've blocked them (put them on my ignore list). However, I didn't provide the names of any training providers that performed inflight shutdowns. I did address one training organization invoked by one of the blocked posters, however (FSI), as has con-pilot, and when they did live inflight training, they did do shutdowns.
Every operator with whom I've flown, and every training department with which I've flown (there have been many), including private, corporate, government, training organizations, etc, that has provided inflight training, has performed engine shutdowns. Every single one.
I recently performed a test flight in the airplane I'm currently flying (a fairly large turbofan multi-engine airplane), in which each engine was shut down, one at a time, in flight, and restarted. Upon returning to the ramp, the final shutdown was performed using the fire handles as a function of the test flight, which was a maintenance test operation. Test of the fire handles to shut down the airplane, stop the hydraulic flow, and interrupt pneumatic flow and electrical power, was a necessary requirement as part of the test hop, and is regularly done. It does not harm the engine, did not harm the engine, and was part of the act of returning the aircraft to service following maintenance.
As previously stated, I've performed inflight engine shutdowns in conjuntion with training in nearly every type of engine I have operated or worked on...every single one an acceptable and approved operation, including required shutdowns during FAA checkrides. It's something I've required as a check airman, and that's been required of me to require, by the FAA, when conducting a checkride.
The pilot who is afraid to shut down an engine in flight, particularly an engine he thinks is "making metal," for fear of hurting the fuel pump, is very poorly trained, and an idiot. Particularly in light of the fact that the manufacturer approves the shutdown, it's a routine procedure, it won't hurt the engine, and it's the appropriate action for the problem the poster believed had occurred.