One certainly can drive in snowy, wintery conditions without snow tires. It is, however, considerably less safe to do so. The same car with the same driver would, in every case, be better off - as in safer to him/herself and others on the road - with winter tires.
Snow isn’t really the problem. Ice is. Winters in the part of Germany where I live are kind mixed up. It can get way below zero and then a couple of days later warm up above freezing - only to freeze solid the next day. I can drive on snow just fine with all weather tires, or even summer tires if I have to - I don’t like to because then I have to be extra careful all the time, and I’m only human so I’ll screw up at some point.
Winter tires are the law here in Germany. You can get a bitching big fine for not having snow tires when needed, and if you cause an accident in the winter and don’t have snow tires then your insurance company likely will not have to cover you - you get to pay all the damages out of your own pocket for being grossly negligent.
To each their own; I drove with all-seasons in prairie winter conditions for decades, and now that I’ve switched to winter tires, I’ll never go back.
I used to think all-seasons were fine until I tried some real snow tires. They really do work better not only at starting up but more importantly at stopping. Around here (Colorado) half the cars on the road are all wheel drive. Yea, all wheel drive enables you to start moving more quickly but doesn’t help you stop at all. Try some snow tires on a road with a thin film of ice on it and you will understand.
How much do snow tires cost? - How much does an accident cost?
Electronic stability control / anti-skid control, ABS, AWD or 4WD High and modern snows all bring a lot to making winter driving safer. I’m impressed at how greatly automobile technology has improved over the years.
We started putting snow tires on our AWD vehicle a couple of years ago. The car/sedan stays with all season tires, mostly for the better fuel economy, but the AWD is what gets driven if the roads or weather are at all iffy. Given the number of 300-400 mile roadtrips we take, I prefer to not be one of the multitude of AWD/4WD vehicles we see in the ditches during a snowstorm.
No, that isht costs too much money. And I believe it is illegal to use snow tires in my state.
Studded snow tires may be illegal; they gouge up roadways. What we’re talking about are winter tires with a tread and rubber compound designed specifically for snow and ice. They’re not illegal anywhere.
I agree that driver skill and sensible driving is by far the most important criterion in keeping the shiny side up, and I’ve lived in Colorado and Michigan combined for almost 30 years, in addition to spending three years driving over the road in a semi in the late 1980s.
Plus I actually prefer RWD vehicles; maybe because that’s what I’m used to. Especially combined with a manual transmission. 
However, if technology exists (ie good snow tires) that gives one better odds when on crappy roads, surrounded by crappy drivers, it’s not a bad idea if you can afford it.
Mind you I’m in SE Michigan; we don’t get nearly as much snow as the northern and western parts of the state, plus it’s pretty much flat as a pancake here. Living and working in a major metropolitan area means that road crews are pretty decent about clearing roads. So, an off the top of my head pulled-out-of-my-butt guess: I drive maybe 20,000 a year. And maybe…one or at most two thousand of those miles are on icy or snow-packed roads. So I can see why some people don’t see the point in spending the extra on tires and rims.
And then there’s this.
I may revisit the notion of getting a spare set of wheels and winter tires. Especially since my primarily-driven vehicle is really not new enough to warrant full-coverage insurance.
I’m not at all hesitant about winter driving, in fact I enjoy it. Combat driving! Yay! 
HOWEVER it’s sort of a downer knowing that if my van gets cracked up whether it’s my fault or the other driver’s*, chances are I will be reimbursed between nothing to little, and I’m out a vehicle. It does sort of change your perspective.
*My insurance agent tells me that in this region, only about 20 percent of drivers keep insurance year around. Meaning 80 percent buy a week’s or a month’s worth right before they get their annual registration, then let it lapse until the following year.
Those things don’t help you stop faster or turn sharper; only good traction (such as is provided by snow tires) can do that.
Except Spain, apparently. 
(See Nava’s post upthread.)
Out of curiosity, where is this?
Genesee county, Michigan.