The computer system is designed with the concept that only 1 subsystem will fail at a time. It determines this by comparing the outputs and using the output that at least 2 of 3 agree (within some tolerance) on. In the case of the ice accumulation, one pitot tube plugged - that left 2 in agreement. Then another pitot tube plugged leaving all 3 in disagreement as the stored pressure would have depleted at different rates due to the moisture bleed port.
That's what triggered the change in control law - the system had 3 different opinions on what the dynamic pressure reading was and there was no winner.
I think the stall came about because when the PIC was handed control he slightly pulled the sidestick back while trying to maintain roll orientation. At that altitude and airspeed the plane was close to stalling and the slight mishandling pushed it to give the warning - the pilot then did what would have been fine if the control law was "normal" and gave it full throttle (causing a hard pitch up) and pulled back, confident that the alpha protection would prevent the stall that was already occurring.
Once the altimeter started unwinding I expect that, even if it was noticed, it would take remembering "how an airplane flies" to convince the pilot to shove the nose down and increase the rate of descent in order to exit the stall and resume normal level flight. I think that at some point the pitot tubes cleared and if they had reset the flight computer it would have recovered hands off because enough valid data would be available. To do that would require knowing that the control law had changed - which the PIC did not appear to know.