At least for my part, I'm in neither "camp". It's a false dichotomy that is a manifestation of the problem.
I'm happy to be in a piston aircraft in which engine failures are simulated by pulling the throttle, provided that that happens to be the least risky way to simulate failures on that specific aircraft with that specific fuel system and engine.
I'm also happy to be in a piston aircraft in which engine failures are simulated by pulling the mixture, provided that that happens to be the least risky way to simulate failures on that specific aircraft with that specific fuel system and engine.
I'm never happy to be in a piston aircraft in which the instructor or ATO is chock full of one-size-fits all rules.
That even regulatory authorities can suggest that one method is generally less risky than another just demonstrates that old wives' tales, like weeds, take hold everywhere. Appeals to authority never impress me.