Captain only approaches
From my experience, captain only approaches are combination of experience and economics. New FOs on the line need to build up experience and therefore are not (themselves) confident to carry out CAT 3 airport approaches until they have gained some reasonable time on the aircraft.
From the economic point of view, captains need to be signed off by a training captain to CAT 3 airports after a number of approaches. As training captains are an expensive resource the purse usually stretches only as far captains to train them.
FOs can carry out monitored approaches to CAT 3 airports, so can fly down to a pre-determined hand over point that has been briefed previously.
As to CAT II and CAT III approaches, again, talking from my experience, they are flown by the FO whilst the captain looks out in order to pick up early visual clues in the low visibility conditions in order to be able to take control of the aircraft at the decision point.
If no contact is established then the FO flies the go-around.