Beagle
I aspire to teaching all my students good operating techniques that apply to all airplanes not just "single engine spamcans". Part of that is when appropriate to the circumstances utilize an emergency checklist to verify you have not missed anything and then complete any remaining items. So for example if the engine failed on the takeoff run I would expect the student to execute the flow by memory and then when the airplane had stopped to go to the emergency checklist and complete the remaining items.That is how professional pilots in large aircraft do it. I see no reason why it should be
any different for PPL's. With respect to an engine fire on takeoff, stopping the aircraft is the priority and the memory items will cause that to happen.... at that stage vacating the passengers and then the pilot is the only additional item and is so obvious that it is not written down anywhere. Since this is a engine failure procedure as well as engine fire drill all of the emergency checks relate to the failure scenario.
My experience is if you ask the average PPL who is one year (or even a few months) after their PPL flight test "what would you do if the engine failed at 500 ft after takeoff ?", you would get a long pause with lots of umms and errrs. Pilots are not born being ready to deal with an EFATO, it has to be taught, and cramming before the flight test may get you through the ride but it will not do much to prepare you for the rest of your flying career. Taking one minute to brief and then practice the actions before every takeoff on every one of your PPL training flights will burn the actions into your brain forever.
However I like to think that my students are the least likely to actually have to deal with the EFATO scenario because I teach them to pay attention to what really matters on the walk around, teach real world fuel planning, and how to do a proper run up including understanding what the engine is actually telling you as well as ensuring the engine is delivering full static RPM at the beginning of the takeoff run..... or in other words eliminate the pilot induced engine failures caused by ignorance or "It is only a spamcan" hubris