I'm pretty sure that is incorrect. What is the point in roll out guidance for Cat IIIB if there are only two autopilots, limiting it to Cat IIIA?
Why would 2 autopilots limit it to CAT IIIa? Two are plenty, the A320 series never had more and was CAT IIIb approved out of the box from the start. As is the 737NG since 2001. We fly to the normal 75m RVR no DH minimum on both types.
To be honest i can't really confirm your story about bad landings and handling on the NG. Yes, it was a change transitioning from the classic and i do prefer the classics handling, especially in roll, but the NG was after some getting used to as easy to fly and land in my opinion.