Icarus. You may have a point re the patronising bit - so I will try and be more concise.
Most multi-engine light twin instructors will agree that cutting an engine at low speeds and close to the ground is risky. How risky is a matter of personal opinion.
It is interesting to note that the British CAA are against simulated engine failures by cutting the mixture of fuel at any altitude (AIC/1999 dated 6th May 1999.
In addition the NTSB in 1976, issued an urgent warning on simulated engine-out manoeuvres which was published following a fatal accident where the instructor and student were killed after a mixture control was used to shut down an engine. An interesting paragraph included in the NTSB warning, could well apply to your own stated reasons for such a practice on take off.
It stated that "Many flight instructors down through the years used the technique of abruptly cutting an engine with a multi-engine candidate to test his emotional reaction and judgement with this extreme technique."
Finally the Piper Seminole Information Manual at Section 10, entitled Training Tips, states that "experience has shown that the training advantage gained by pulling the mixture control or turning off the fuel to simulate engine failure at low altitude is not worth the risk assumed - therefore it is recommended that instead of using either of these procedures to simulate loss of power at low altitude, the throttle be retarded slowly to idle position".
If you feel so strongly about using the mixture as a good training policy then it might be safer to do this above circuit altitude where there is more time to recover from stuff-ups by either yourself or the hapless student. If the purpose of the map between the throttles and mixture trick is to give practice at identification then why not practice this above 2000 ft until the pilot is competent at correct and speedy identification.
Having now demonstrated his competence you can then use the throttle method at lower altitudes which is demonstrably a safer technique. But as I said earlier the policy of bunting over to land straight ahead is really beyond the pale for any sensible instructor.
All this reminds me of that wonderful pithy saying coined by some
wise aviator of yesteryear:
"A superior pilot is one who stays out of trouble by using his superior judgement to avoid situations that might require the use of his superior skills". A lesson there for you perhaps, Icarus?