Hi again,
I managed to enable the interface by sending a tap event through the adb shell ("input tap 1000 100"), which brought the tunnel editor, from which it is possible to navigate to the enable button.
And the answer is... It works!
I could verify that, when using a browser on the Fire TV, my public address was the one of my VPN server. Quite handy if you want to take your stick on holidays and still be able to watch content from your home country, for example.
This is obviously hacky. In order to be usable, the app would need to handle the navigation key events. I noticed that It already handles KEYCODE_BACK, so I guess this could go together. Amazon has a section about it on their documentation [1].
Cheers,
Christophe-Marie