BE-76 Flight Instructor reported incorrect procedures were used during flight training and resulted in an engine-out landing at departure airport.

Narrative

The practice of shutting engines down on pilot examinations should be re-evaluated by the FAA. There are two methods to restore engine operation inflight after an intentional engine shutdown in the Duchess. One method uses an unfeathering accumulator and the other relies on the engine being started with the propeller still in feather. Both are viable methods. Name is requiring Name 1 to retest this portion of the examination because he failed to properly execute the checklists. This explains to me why the unfeathering accumulator procedure was unsuccessful for Name 1 and Name. The accumulator has only one charge of oil to unfeather the propeller and if it is improperly applied; it will be unsuccessful at unfeathering the propeller. The procedure then requires the engine to be started in feather.Name 1 again failed to properly follow the checklist; leaving the mixture at idle / cutoff while cranking. At this point Name took over control of the Duchess. If Name 1 and Name had properly performed the start procedures in the air; we probably would not be having this conversation. Allowing a deliberate engine-out situation to become unrecoverable and having to [request priority handling] for an engine-out landing is undesirable. I would not allow another pilot to incorrectly proceed with needless cranking of an engine with its fuel cut off; and would have intervened sooner to avoid a potentially overheated starter. The procedure requires the starter to be held engaged until the propeller starts to windmill. The procedure also does not instruct a pilot to dive the aircraft or perform any banking maneuvers. I believe the practice of intentionally shutting down engines on check rides; coupled with operator error; is a liability for safety.To clarify; the actions taken on Date are in compliance with Beechcraft Duchess normal procedures. After Name 1 and Name were unable to start the engine inflight; it was a valuable instructional activity to observe Name 1 start the engine using normal procedures.First; the BE-76 POH Section II; LIMITATIONS" does not contain a limitation that prohibits starting an engine with a feathered propeller. I consulted Name 2 of Company Maintenance and he confirmed the engine may be started with a propeller in or out of feather. We discussed how the propeller was in feather because the previous pilot had intentionally secured the propeller in feather during flight and not a system malfunction.Second; I supervised the preflight in which the POH requires the propeller to be examined for nicks; security; and oil leaks; which was performed. Third; the engine was started per the POH with the propeller lever HIGH RPM. The engine started with the ship's batteries within the engine starter duty cycle. Finally; I supervised the Before-Takeoff Checklist; where all systems performed normally."