Swash,
As a matter of fact it is not only the aircraft type that does it. The type is where it starts. My 737-3/400s are CAT IIIA. this was at first a surprise to me but it works OK.
Now this stuff needs maintenance so the local authorities are supposed to keep a close eye on the maint procedures.
Once you have a properly equipped and maintained aircraft you need CAT III trained pilots. This is a training and experience thing on top of the standard IF rating.
Of course you can't autoland if you don't have a proper ILS. Similarly you can't fly CAT I, II, IIIA, B or C if the ILS is not at least qualified accordingly.
That is the chain. One link breaks and you are back redoing your fuel calculations.