The booster device isn't an amplifier. It's called a boost converter or boost regulator. I thought that you were talking about an audio amplifier.
The boost device is fine. It will give you a larger voltage.
However, without knowing what sort of motor drives the fan, you can't be sure whether it will go faster, or not be damaged, if the voltage is increased.
If it is a simple DC motor, a larger voltage will make it go faster, but it will increase the heating of the motor. Do you have details of the motor?
The circuit looks about right, but I'm not sure what the delay on the relay does. You might well be able to get away with leaving the output of the boost converter permanently connected to the fan, and a diode to stop current flowing back into the normal fan controller.