Prior to doing any failures that involve an engine shutdown & feather I checked to see if the a/c would at climb with one idling.
If it would climb with a windmilling prop then I had some reassurance that it would do so with it feathered.
When I've been instructing I've also seen (& experienced myself) the "The instructor's hand moved - here comes/there is the engine failure" syndrome. At altitude I also used the fuel selectors.
Amazing how many students during their troubleshooting talk about & touch the fuel selectors but don't effect a change in the selector's position - even when the fuel selector is the cause of the stoppage.
I had one bloke go straight to feather because his 'checks' didn't restore power. It was in the cruise & had planned to give him shutdown anyway. It was only while securing the engine he noticed the fuel selector.
Another fellow did something similar. He chose to feather as well then afterwards decided to to his troubleshoot checks. His initial procedures were more appropriate to an after t/o failure.
In both cases during the flight the students had previously correctly handled a failure using throttle. I put their less successful efforts down to a lack of the 'prewarning' that is gained when the student sees the instructors hand move to the throttle quadrant.
As for using mixture for cuts, I do above 500'. The engine is still windmilling, there is still spark & air occuring. Mixture can be advanced to restore power.
I can hide both mixtures so the student must use something other than throttle position to determine the failure, and can also test his/her initial determination by closing the suspect throttle.
After doing the drills I set zero thrust. The time period during which the mixture is closed is rather small.
I've never had trouble with with mixture failing to restore power but have had many times when the plugs foul at full rich & idle power.
As an aside, in the UK it's done on throttle and the student is expected to use yaw (& gauges if circumstances permit) then go straight to feather (after t/o situation). No testing of the initial assumption.
I found it quite awkward getting used to that when I converted to the local bit of paper.