The ATO himself/herself, and not the student, has failed the test if mixture or fuel off is used to generate the failure on take-off.
Under any analysis, that's putting the aircraft at significant risk.
I don't even know why this discussion is still occurring.
The discussion is still occurring because there are still people who'd like matters of aviation safety to be determined by data and rational decision-making that takes the data and relative risks into account.
I can show you a certified and confirmed serviceable aircraft (including a confirmed serviceable fuel system) whose engine will continue to run even with the throttle closed and the mixture set to idle cut-off. How can that be so?
On "any"
evidence-based "analysis", pulling the mixture control does not have the effect of turning the "fuel off" on
many aircraft. On "any"
evidence-based "analysis", on
some aircraft the simulation of an engine failure by pulling the throttle produces an operational environment that is unrealistic of the usual causes of engine failures.
Don't you think it would be safer if the method of simulation was decided on the basis of an understanding of the particular systems on the aircraft in which the simulation is occurring, and a consequent understanding of which method is - on balance - least unsafe?