Try as I may I cannot understand your statement that the mixture cut does not shut down the engine.
The propeller has not been feathered, there is still a source of ignition. Enrichen the mixture on your standard twin trainer, i.e. the Duchess, and power will be restored in less than a second. Doesn't sound shut down to me.
Due to there being no risk in using the mixture to simulate the engine failure, lets consider the other benefits of using this technique. The trainee now gets the benefit of using the throttle to verify the failed engine (i.e. check that changing the throttle setting doesn't cause any yaw). They also get to practice closing the 'failed' throttle and using it as a flag when shutting down other systems. You can't get those training benefits if the instructor closes the throttle.