i think we have to differ between a training and precautionary shutdown.
and i think we basicly can agree that a shutdown and restart of an healthy engine inflight is technically possible, on pistons, turboprops, big turbofans.
a precautionary shutdown is of course always pilots decision and there is not much to discuss.
a training shutdown, and here i can speak just for my experiece as well european rules, is not allowed. i am not sure how it is regulated e.g in the USA.
the insurance simply will not cover such a thing.
in europe you need a typerating even on smaller multiturbine planes , and here you do not have many simulators for that this so you often perform the rating on type. on the conquest II we have put one engine just in idle since fully feathering a garrett means also to shut it . the NTS system can be checked on ground and we never tested it inflight
on the cheyenne III we feathered the prop but kept the engine running ( the two shaft free turbine design makes it possible)
on the BAE146 in my airline time all such things were performed in a simulator in manchester and on the real plane we never put an engine even in idle for training purposes. the line checks cover normal procedures as well theoretical knowledge- for the rest go into a simulator.
you may find people who shut "illegally" at small turboprops for training purposes- or maybe in other regions there other rules and behaviours, so maybe some statements above are correct. the military may also have other rules.
but- in nowadays forget any story written above that on a line check anybody shuts intentionally an engine at a big airliner by e.g pulling the fire handle , that are laughful stories. i bet you will not find one video at youtube where lets say in a real 777 or 747 the checker pulls the fire handle on a line check. it would be his last.
i am wondering that anybody who really earns his money with aviation believes that "check airman" stories written above instead of laughing about it. again- try in nowadays at united airlines to shut inflight intentionally a turbine of a widebody airliner for training ... show me that captain or line checker...
on the BAE146 we from time to time put on ground and the engine off the fire handle to its first detent to check the bleed air valve trips and the fuel supply also. pulling the second detent of course would mean to fire the bottle... thats not pretty.
when it comes to "hurt" a turbine by an inflight shut... well, in my opinion its nor dramatic neither healthy. on a singleshaft turbine design ( tpe331, T56) all turning things inside the engine stand still when the prop does and all is fine. in a multishaft turboprop or a turbofan you have a windmilling . here you may be restricted to a windmilling without proper lubrication.
on pistons you also have to consider the shock cooling of the cylinderheads.
cheers !