> I'd heard the folklore that snow tires are more important than all wheel drive for handling on both snow and ice, and having used them for a few years now, I'm a believer.

That seems obvious. AWD means you can use the traction of all tyres to move. But without snow tyres you have no traction, so AWD does nothing.

> So called "all season" tires are not snow tires.

Not quite true, there are 3PMSF all-season tyres, and unrated ASTs might behave well on snow, it depends on the compromises the manufacturer decided on, as the point of an ASF is to balance dry, wet, and snow. But where the manufacturer decides to put the balance is very relevant. That a tyre is all-season doesn't mean it's all-of-your-location-seasons.

In fact, you can have ASTs which are excellent in the snow and absolute shit on dry and wet, that's what "tyre reviews" found with the Tomket Allyear 3 (https://www.tyrereviews.com/Tyre/Tomket/Allyear-3.htm) for instance.

Also technically unrated "winter tyres" may not be suitable for snow either, they might be using better winterised compounds and deeper thread patterns but not sufficiently so to behave well on snow.