No instructor in his right mind would seriously consider cutting a mixture in a single engine trainer after take off. Well, why not? The answer is simply because 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.
I'm not sure I agree entirely. I cannot answer why engine failures are done this way in singles, except for the fact that Cessna singles like the 152 and 172 have a dirty habit of breaking mixture cables at times.
To me there is a lot of sense to simulate engine failures on singles by "Pulling the mixture". So often when the engine is asked to respond at the end of the exercise, even with normal engine warming/clearing procedures, it can be slow to respond or run rough usually due to plug fouling, something that cannot happen with a lean mixture. Also the chance of carb icing is reduced because there can be no evaporative cooling taking place.
So long as the engine is windmilling all that is needed to restart is advance the mixture.
There will be exceptions but I don't think there is any more risk in using the mixture in a light twin than the throttle to simulate engine failure. One big thing for me in multi engine training is, as has been stated by others, that the mixture cut most closely simulates a real failure and confirmation cannot be obtained from the MP or RPM guages. EGT and fuel flow need to be checked.
I agree entirely with you about the stupidity of simulated engine failures at low level.