Now that I have received the module, I have added it to the circuit accordingly to the scheme. The LED is blinking when the module is on and stays on when a device is connected.
At first I forgot to solder a simple wire to the "EN" pin (for enabled I presume), it simply did not work... After correction, it worked pretty well! Next steps are to deal with volume pins.
To supply 12V power to the stereo amplifier, we worked on a little device that deliver it with to AA batteries. We are going to write another hackaday project for this one.