Hello,
So, I've been living in the Swiss Alps, near the Austrian border, for over 11 years.
Every day, I was on the road, facing all kinds of weather. I had never
experienced a situation where I needed snow chains.
As long as you don't have to drive up remote mountain roads just to pass through, I think it's completely unnecessary.
Main roads are usually very well cleared, and no one forces you to use obscure mountain roads. Just stay on the highway and use good winter tires.
Best regards, Rainer.