Think about a scenario where there was no tilt at all of the earth's axis.
There would be no seasons anywhere on the planet (a slight change in our summer since the earth is slightly farther from the sun.)
But at the poles, it would just be constant sunset all the time, 24 hours a day every day. With the thicker atmosphere that the little amount of sunlight has to go through, it would be 40, 50 degrees below zero all the time, 24 hours a day, day in day out.
There would be no melting of any snow that falls at the poles. Massive glacial sheets would build up, flow south (and north) and the top third and bottom third of the planet would be just one big glacial sheet.
Just the tilt moving from 24 degrees to 22.5 degrees can leave an effect which could be similar (when combined with other cycles.)
Just having a continent over the poles like Antarctic and Greenland/Baffin Island can cause glaciers to build up. 600 million years ago, most of the continents were locked up together over the south pole and nearly the entire planet froze over (even the oceans.)
Like I said, none of this has ever been taught to any climatologist.
"There would be no melting of any snow that falls at the poles."
If there were no evaporation at the poles there would be no clouds from which snow could fall.