Ideally the stall should be detected at the incipient stage, in which case the stall recovery you described would be correct. However once the aircraft has stalled you don’t want to use aileron as the controls may well work in the reverse sense. As for pushing the control column forward, no, it’s just meant to be sufficient to unstall the wing, and in most training aircraft this will only be a couple of centimetres. The last thing you want is to pole the control column all the way forward.
It would seem that the FAA has moved on from that theory too, now their focus is on properly breaking the stall rather than the height loss aspect.