In the mid 60s we would shut the engine down and feather the prop at 300 feet after takeoff in a twin Dart airplane for asy practice. Made lots of actual single engine landings and go arounds. We had the engine that was shut down ready for a relight. I had an actual engine failure at 500 feet once with the other engine shut down and got it running in time, but it was exciting for a while.
Had 5 engine problems requiring shut down in that airplane and the experience gained in training was invaluable. Made them non-events.
The policy was changed to only allowing simulated shut downs but not for any problem we might have had, just that it was felt to be safer.
I did appreciate the experience and think that I know a lot more about single engine flight because I had that opportunity. The point about the landing swing is a good one and there is no way to simulate that, even by leaving the failed engine at high idle. The yaw with an engine feathered is very dynamic.
I have had problems a couple of times with setting the wrong simulated power, in that the prop drag was worse than a feathered prop would have been and the airplane was on its way down, requiring quick action to save it. Usually turboprop engines that have a big prop drag, such as Garrets.
Twice have had a student roll almost inverted due to Vmca during training with the failed engine at idle, so I am familiar with the risks. My first twin was the DC3 and that airplane really teaches you a lot.
I shut an engine down in an Aztec for training and could not get it running again due to battery failure. Generator on good engine failed too, so we had no electrical power and sun went down before we could land. No radio etc luckily runway lights were already on. Of course the running engine did not have the hyd pump. It was a great learning experience for the student. Most pilots never have a chance to do this in their entire flying career.
You could not buy that experience. But in retrospect I think using the throttle to simulate an engine failure is best.