Here's what I did:
1) Removed the bracket and pulled the GoPro out of the socket
2) Pulled the battery and put it back in again
3) Inserted the GoPro into the gimbal and tightened the bracket to fix the GoPro
4) Switched on the Phantom and et voila, it worked! The gimbal switched on & charged the GoPro!
---------

------------ but wait...
5) Switched off the Phantom, but the GoPro stayed switched on
6.1) Switched off the GoPro, then I switched on the Phantom but the GoPro stayed off & didn't charge anymore :|
6.2) Switched on the Phantom with the GoPro still turned on, but still the gimbal didn't charge the GoPro anymore
==> To conclude. In order to make the gimbal charge the GoPro, I have to unmount it, remove the battery, install everything again and then it works exactly
one time. After that, I have to remount everything all over again in order to make the gimbal charge the battery.
Obviously this is not how it's supposed to work. What could be the reason for that? Gimbal broken or GoPro broken?