Well there is, as they say, more than one way of "skinning a cat" but if we want to encourage "thinking" pilots it would rather depend on the nature of the failure.
If there has been a big bang and there is a conrod sticking through the cowling there aint much mileage in checking round for cause of failure.
If the enging has simply run down and the prop is still windmilling then perhaps it's fuel in which case put the pump on and change tanks (irrespective of indicated contents), maybe also check the mixture is rich.
If it's rough running/partial loss of power then perhaps it's carb ice or a u/s mag etc.
But in reality its also a question of time and priorities. If you are at FL 100 when the engine fails you might have many more options and perhaps plenty of time to do some troubleshooting and try a restart.
On the other hand, if you are at 1,500 agl you may not have any time to do a trouble finding check - the number one priority in any a/c is to maintain safe control and concentrate in getting into you planned field etc.
I am not against having a methodical approach to this issue but I think the question is a bit loaded towards the wrong priorities in the event of an engine failure.
Finally, I often initiate practice forced landings by advising the student he has a rough running engine and/or throttle back to 2,000 rpm say and tell him/her that this is the maximum revs available rather than just closing the throttle completely. I think this approach tends to get them thinking about the issues of engine failure more deeply.