> I'd heard the folklore that snow tires are more important than all wheel drive for handling on both snow and ice
All wheel drive helps zero with winter conditions, except for going uphill or some other acceleration-equivalent action. Quite the opposite really - many people have experienced how with 4x4 you don't get a warning before you just lose your grip and off the road you go. For the record I'm driving a 4x4, but using summer tires in the winter, with or without engaging 4x4, the car will simply be like Bambi on the ice. Going downhill would mean that I wouldn't stop before hitting something or ending up in the sea. 4x4 doesn't matter for that.
The first thing I do when getting a new car (I always by second hand cars - not that I buy them often, but I'm old enough to have gone through a few) is to buy the best possible summer and winter tires I can get, particularly the latter. That goes before any other investment in the car. I had enough of bad tires when I was very young and without any money.
EditAdd: Many people comment about ice and studded tires. Yes, studded tires are better, not just on ice, but on many kinds of packed snow conditions as well. For some conditions they are the only good option (interestingly enough, this does not always mean that they're better than studless tires on all kinds of ice - it depends on the ice). However, if there's a lot of bare asphalt around, as is often the case when there's a lot of traffic, then studs aren't particularly popular (road wear etc), and in some countries in Europe with lots of bare roads in the winter they're illegal (e.g. Germany IIRC).
As for winter tires without studs, on ice, be aware that there are great differences between brands and types. My father's car had stud-less winter tires while my car had studded tires, and I drove both cars on the same roads and his car behaved well, even on ice or "glassy" snow. But then the tires got old and he replaced with brand new tires, but a different type - much newer, so should be better, right? Not so. Very different to drive. Dangerous I would say, and in fact he got into an accident because of that. I checked that place with my own car afterwards, and no doubt his old tires would have held up just fine.
