The *airport* must also have a CAT II/III rating. Not every runway in the world is capable of these landings, only a very small percentage. The CAT II runways need inner marker equipment and need their ILS beams tuned to much finer tolerances, and the CAT III ones have even more exacting standards.
A little old DC-9 that does runs to Nowhereville all day doesn't need a CATIII rating because most of the airports that it serves won't have CAT III runways. And even if the airport is CAT III rated, there's not a big problem in delaying the arrival into Nowhereville for a few minutes to wait for the fog to lift.
On the bigger planes, it becomes more important. They'll spend all their time around bigger CAT III airports. A big 777 arriving at a hub airport with 300 passengers, many of whom are connecting to other flights, MUST get there on-time or else it would throw the entire airline's schedule out the window.
A pretty good percentage of the old commercial planes out there today, meaning the old 737s and 727s, are at least CAT II rated, which is enough to keep the airline's schedule running smoothly in case of fog.