All that is necessary is for some melt water to get into crevasses, which are lines of weakness in the ice shelves. Surface melting exploits the line of weakness, and the surrounding area of the ice sheet can disintegrate without having gone through a longer term melting process. The snow line, above which precipitation falls as snow, intersects the shelves at a very low angle. This is what happened to the Wordie Ice Shelf on the Antarctic Peninsula. Just a little bit of warming during the summer was enough to change the whole pattern of snow buildup in winter followed by melting in summer.
Sea ice is also full of cracks, and so it can disintegrate before it melts.
In the long winter night in the Arctic, there is black body radiation. But even at night, part of it can be reflected back to Earth by clouds, and if there is an increase in cloud cover, this too would tend to increase warming. Removing sea ice increases ocean evaporation and therefore should increase cloudiness in the Arctic and Antarctic. Farther out the permafrost is thinner and may be subject to quite a bit of melting in the summer. By and large it is not subject to sudden change.