As mentioned there is a distinct difference in limitations across the Airbus types. In particular between the narrow body and wide body fleets.
To summarise
Airbus is not in a position to recommend autoland on A320 or A321 in Overweight.
In order to understand the Airbus position regarding the difference between A320, A321 and other types such as the A330.
The A330 Overweight Landing procedure in FCOM PRO-ABN-80 starts with the following information:
"Automatic landing is certified up to the Maximum Landing Weight (MLW). Autoland flight tests have, however, been successful up to 229 t (504 853 lb). Depending on the situation (e.g. emergency or other) and provided that the runway is approved for automatic landing, the flight crew can decide to perform an autoland up to 229 t (504 853 lb)."
The A320 and A321 Overweight Landing procedure does not provide similar information because no flight tests were performed to support this information.
The purpose of this information is to ease decision making in case of overweight landing. In general, autoland is beneficial in order to reduce the flight crew workload.
Starting with A319 certification, Airbus took the opportunity of autoland certification campaign to fly some additional autoland up to the MTOW.
However, the reduced quantity of tests does not enable Airbus to publish certified data. Without these flight tests, Airbus is not in a position to recommend autoland on A320 in overweight.
The use of autoland will be a decision of the flight crew, depending on the weather conditions, of an emergency situation, and of the flight crew workload.
Hope that helps explain the FCOM wording.