I would think that, with a roof that is burning about 25 meters above the floor of the cathedral, most of the damage at ground level would indeed only be from debris that fell down, not from the heat or fire itself. Heat still has that tendency to rise, fortunately.
Also, with most of the ground floor way more accessible to the firefighters than the roof itself, it seems entirely possible that they were able put out any burning pieces that fell down more easily, further limiting damage at floor level.
Indeed, the real problem with putting out the flames was probably the logistical (and physical) challenge of pumping enough water up to roof height!
