Winter tyres are awesome, they work incredibly well.
Snow chains suck harder than a Dyson plugged straight to the national grid.
Seriously, we live in a ski resort in the French Alps and life would be pretty unbearable without winter tyres. Chains are hopeless. Fitting them is a total faff which inevitably leaves you cold and wet. They do work well once they're on, but they're very fragile (we memorably trashed 3 sets in one weekend last December when it was puking snow non-stop for days on end).
With winter tyres, you might, in a real worst-case scenario, eventually still have to resort to chains, but this is usually avoidable with some pre-planning and a bit of driving skill. Just outside our chalet, there is a steep slope on the road. It's almost always covered in bullet-proof ice as we're above the resort centre and the snow-ploughs can't be bothered doing much. We never have any problems getting up it unless we have to stop part-way for some reason (usually muppets walking up the middle of the road in blissful ignorance of the revving, hooting vehicle behind them).