While I definitely agree that the priority in
any emergency including an engine failure is to fly the aircraft, I don't agree with the premise that a full cause check should not be be done in most circumstances. So for example if the reason the engine failed was because you had the mixture leaned for low cruise, descended and then added a bunch of power causing the engine failure, you are implying it would be better to go ahead and crash rather then perform a 10 second cause check which would instantly restore full power
My Mount Doom example - engine lost power, instructor set up glide to a useful piece of water (we were on floats), then performed the cause checks ... and we both simultaneously spotted that the mixture knob was further out than either of us had deliberately put it, problem solved. From beginning to end: maybe ninety seconds. (I think his knee had pulled it out a bit when we hit some turbulence whilst he was turned round talking to my family in the back.)