If the `failed` engine does not pick up when the mixture is moved from cut-off to rich then the situation becomes grim indeed as that engine is now dead and the prop windmilling.
the aircraft would be left with no power if the mixture was moved from cut-off to rich and nothing happened for whatever technical reason
Sorry A37575, your argument is not specific to mixture cuts. If you use throttle to simulate engine failure and the engine doesn't pick up when you move the lever back up, then you still have no useful power. The mechanics of the throttle linkage and the mixture linkage are very similar, and neither is any more likely to fail than the other. Power is returned just as quickly whether you use throttle or mixture in the simulated failure. The gamble (as you put it) is exactly the same one, with exactly the same odds of losing.
There are two things that do give me the heebees in asymmetric training - Simulated failures at low level, and using fuel shut-off valves to do the failure. Low level failures have been thoroughly thrashed here and elsewhere, and using fuel valves to stop the fuel means an extended and indeterminate delay once fuel is turned back on before fuel will be returned to the engine:
That is the really bad choice in all of this.