“ice sheets that may have been several miles thick crept over every inch of Earth’s surface.”
How would that work? As the ice sheets grew larger and larger, there was less and less ocean surface for evaporation needed to create snowfall. When the ice sheets covered half of the ocean surface, evaporation would be reduced by half and snowstorms would be greatly diminished.
One would think that would be a self-limiting phenomenon, that you could not get to more than 50% or 75% of ice coverage of the oceans before there simply would not be enough snowfall to continue glacial growth.
But the glacial the ice sublimates (goes directly from solid to gas without passing through the liquid phase). So the water cycle could continue due to atmospheric water originating from sublimating glacial ice rather than evaporating ocean water.
In the case of glaciers, sublimation can occur when the air temperature is below freezing and the air is very dry (as the oceans got covered in ice, the air would become drier and drier). Under these conditions, ice at the surface of a glacier can turn directly into water vapor and be carried away by the wind. This process can occur even when the air temperature is below freezing because the heat required to sublimate the ice comes from the ice itself, which cools as it sublimates and can keep the surface below the air temperature.
One theory is the Sun goes into a ‘cool’ period and doesn’t provide enough energy to the Earth to keep the oceans from freezing............