Am I the only one here thinking that the climb was not induced by the pilots but by some sort of ill functioning protection?
No - you yourself provide the quote from the BEA that indicate it was initiated by the crew (my bold):
From 2 h 10 min 05, the autopilot then auto-thrust disengaged and the PF said "I have the controls". The airplane began to roll to the right and the PF made a left nose-up input. The stall warning sounded twice in a row. The recorded parameters show a sharp fall from about 275 kt to 60 kt in the speed displayed on the left primary flight display (PFD), then a few moments
later in the speed displayed on the integrated standby instrument system (ISIS).
Note 1: Only the speeds displayed on the left PFD and the ISIS are recorded on the FDR; the speed
displayed on the right side is not recorded.
Note 2: Autopilot and auto-thrust remained disengaged for the rest of the flight.
At 2 h 10 min 16, the PNF said "so, we’ve lost the speeds" then "alternate law […]".
Note 1: The angle of attack is the angle between the airflow and longitudinal axis of the airplane. This information is not presented to pilots.
Note 2 : In alternate or direct law, the angle-of-attack protections are no longer available but a stall warning is triggered when the greatest of the valid angle-of-attack values exceeds a certain threshold.
The airplane’s pitch attitude increased progressively beyond 10 degrees and the plane started to climb. The PF made nose-down control inputs and alternately left and right roll inputs. The vertical speed, which had reached 7,000 ft/min, dropped to 700 ft/min and the roll varied between 12 degrees right and 10 degrees left. The speed displayed on the left side increased
sharply to 215 kt (Mach 0.68). The airplane was then at an altitude of about 37,500 ft and the recorded angle of attack was around 4 degrees.
From 2 h 10 min 50, the PNF tried several times to call the Captain back.
From my read of the above, the pitch up command pre-dates the Stall Warning (though we do not have a well defined timeline to understand exactly the second by second sequencing).
We can see that the initial Pitch Up demand led to the climb, ultimately reading 7000ft/min, prior to the Pitch Down command which slowed the climb rate to 700ft/min, however, it seems by then significant airspeed was lost.
The text that follows in the report seems to be the most perplexing:
At 2 h 10 min 51, the stall warning was triggered again. The thrust levers were positioned in the TO/GA detent and the PF maintained nose-up inputs. The recorded angle of attack, of around 6 degrees at the triggering of the stall warning, continued to increase. The trimmable horizontal stabilizer (THS) passed from 3 to 13 degrees nose-up in about 1 minute and remained in the latter position until the end of the flight.
Around fifteen seconds later, the speed displayed on the ISIS increased sharply towards 185 kt; it was then consistent with the other recorded speed. The PF continued to make nose-up inputs. The airplane’s altitude reached its maximum of about 38,000 ft, its pitch attitude and angle of attack being 16 degrees.
At this point we've seen a half-heart attempt to use pitch-down, which may just have been the solution. But that appears to have been abandoned. Pitch up demands then remain, until very late in the sequence, at which point the vertical speed was already at -10,000ft/min with an angle of attack stated to be +40degrees - the stall was full developed and they ran out of altitude.
Re-reading the report is a sad and sobering task. I simply do not understand the initial pitch-up demand. All that seemed warranted was a correction due to the initial right roll? However there is no mention of any throttle demand at the point where the first pitch up command was provided, so presumably airspeed is bleeding off from this moment on. TO/GA thrust was eventually selected 46 seconds after AP and AT disengagement. Too late.