I think you're confusing two different functions, flight envelope protection and stick push.
A320 series flight protection does not pitch rapidly down when it detects an approaching stall- it maintains the AoA for maximum lift as long as I command it, and (except in the final phases of the landing sequence) if a stall is imminent it applies TOGA thrust to try and go up.
Stick push, on the other hand, does force a pitch down to prevent a stall- but stick push is not limited to fly-by-wire aeroplanes; in fact conventional stick push is not a feature of fly-by-wire systems, as the flight envelope protection prevents the stall being reached.