For what it is worth from the viewpoint of someone who was involved at the time and since in setting operating minima, the BA Tristars (and all the fail operational Cat III types that followed) were never formally operated to IIIC limits as we always imposed a minimum RVR of 75m (100m in the case of the 747-400) which is in the IIIB range in the ICAO definition. This minimum was set after a bruising negotiation with the UK Pilots' union, BALPA, who were worried about the prospects of the Fire Service ever finding an aircraft in trouble in true zero visibility. One could argue at length (and we did) about whether the scenario is particularly realistic (if the damn thing was on fire, finding it would not be a major problem...), but in the end honour was satisfied when it was realised that BA simply didn't need clearance to operate in visibilities that were never observed; for all practical purposes an RVR of 75m is as close to zero as you are going to get and in the 30 odd years since the number of Cat III flights that have been affected by an RVR below limits in BA is very close to zero.
Where I believe the Tristar does deserve a place in history is that it was probably the first to be operated to a genuinely zero DH. Under FAA rules the minima may be described as 'No DH', but I believe this is a legalisitic way of saying that no DH has been specified - a sight of the ground is still needed before touchdown. The UK rules, however, do permit what it says on the tin, no view of the TDZ is needed. On the Tristar insisting on some forward view of the TDZ would have limited the aircraft to about 125m RVR, so high was the pitch attitude on touchdown. The UK CAA agreed to this interpretation after long discussion during which it became clear that no one could really say what the 'decision' on a fail operational Cat III system was for!
The BA Tristars were indeed equipped with Ground Roll Monitors, intended to help both taxi out and identification of runway exits on landing, but the minima did not depend on the GRM being servicable, which was just as well as they weren't terribly reliable. The Trident 3s were also fitted with them, but were always operated to a Cat III DH of 12 ft; the Trident Autoland system, advanced as it was for the time, never inspired quite the same degree of confidence as the Tristar's, usually described as 'PFM'. The RVR limit for the Tridents came down in stages to 75m too, so was also not IIIC.