Lets says they did lose an engine on take off…
Why not simply pull the throttle back (of bad engine ) instead of touching the Fuel Control Switch ?
Because that’s not how jet engines work.
First - you fly the airplane, then, you make certain that you’re on the correct ground track to keep you clear of obstacles, then, and only then, you start working the engine problem.
The independent confirmation of which engine is being shut down is critical. The wrong engine has been shut down before, and it will probably happen again, as a result of flight crew error.
These guys did it.
These guys may have done it.
But, once the correct engine is verified/confirmed, you have to stop pumping fuel into a bad engine. It’s not like your car, you don’t just let off the gas with an engine problem.
This is an uncontained failure - it’s a good idea to shut off the fuel to the engine before (or after) this happens.
Letting a damaged turbine engine run means you’re about to start a fire, or have an uncontained failure, so, no, you don’t just go to idle on a bad engine, you go to idle, see if the problem abates, and then shut it down, first with the fuel control, and then, you pull the fire handle to secure/shut off bleed air, fuel shut off valves, electrics, hydraulics, and arm the fire bottles.
Letting a bad engine run will only increase the damage that it does, to itself, and to other, important, parts of the airplane.