Uncas,
You're right! When you pack a snowball it is harder to break apart. When snow packs into your tire it gives increased traction in snow/ice much like packing a snow ball makes it harder to break apart.
Understanding this idea, a man named John F. Sipe cut grooves into his shoe soles to increase traction on slippery surfaces. His son promoted this benefit in tires. This greatly increased the traction. As tire rubber compounds improved, siping became more prevalent.
"In 1978 the US National Safety Council found siping improved stopping distances by 22 percent, breakaway traction by 65 percent, and rolling traction by 28 percent on glare ice."
The top winter tires all have lots of siping which allows snow to pack in to the grooves aiding traction in snow/icy conditions.
Conversely, in deep snow you have two choices...tall narrow tires with large voids allow the tire to dig down to ground for better traction; or fat flotation tires to ride up on top of deep snow...
Eddie