The point of leaving the engine idling or in the case of a petrol driving with less throttle for the last part of your journey is not to cool the actual body of the turbo. The turbo is designed to run at extreme temperatures. The problem is that the oil we use in our cars is not.
When you turn your engine off, you stop the oil from circulating and the residual oil that is left in the oil pathways slowly runs back into the sump. A very hot turbo will bake and carbonise its residual oil in short order. These carbon particles are baked on to the internals of the turbo and can severely reduce oil circulation on subsequent runs. That is why its important to give the turbo as much help as possible in cooling down. A bit of 100 deg C oil circulating around a 700 deg C turbo for a couple of minutes makes a world of difference.
When you turn your engine off, you stop the oil from circulating and the residual oil that is left in the oil pathways slowly runs back into the sump. A very hot turbo will bake and carbonise its residual oil in short order. These carbon particles are baked on to the internals of the turbo and can severely reduce oil circulation on subsequent runs. That is why its important to give the turbo as much help as possible in cooling down. A bit of 100 deg C oil circulating around a 700 deg C turbo for a couple of minutes makes a world of difference.