FFF
On SE aircraft, I think you have made the same fine point that many others have made.
On ME aircraft, the issue you may have missed is the difficulty in determining that an engine has actually failed at all when on approach, because of the low power settings typically being used, and the resultant lack of significant yaw. Once it is determined that an engine has failed, the recovery actions will depend on how close you are to the field, and whether your position and speed actually require any significant increase in power on the live engine. If little or no increase in power is required, you really don't have to go through the drill of pushing all the levers forward a la EFATO. Sort the problem out on the ground.
The particular problem that I suggested earlier, is that when an engine is operating at low power, it is often only as you apply power to it, that the failure will become obvious. The most likely time for a ME pilot to be applying power on an approach is as the gear and flaps come down, and/or if he sinks below glide. In both cases this is not a good time to discover the loss of an engine.