"How would it be closed due to snow (rather than a lack of snow)? "
David, you should put that question to the National Park Service when they publish this information on the web site.
When the Badger Pass Ski Area is open for business, they pay for having their parking lot plowed. The road is open that far, so downhill skiers use the lot, and cross country skiers use it as well. When Badger Pass is not open for business, they don't pay for plowing the lot. Therefore, there may not be a safe place for cross country skiers to park, so NPS leaves the road to Badger Pass open or closed, depending on its mood. When there is any doubt, they leave it closed.