That’s what practically everybody here does, unless they use all-year tires (which are nowhere near as good in winter conditions as proper winter tires). It’s often said here that the use of winter tires is obligatory, and that is certainly the case when the road conditions dictate: black ice, glazed-over snow, snow-mush, or generally icy conditions. There’s also no fixed period when such tires must be used, but the generally-accepted period for their use is the so-called “O-to-O” period, that is, October to Easter (Ostern). So, since it’s too much hassle to keep changing over sets of wheels when it’s icy/snowy or not, people keep a set on the car for the entire winter, and a set in the garage or cellar for summer. And, when one buys a car second-hand here, one will often find that the car comes with at least eight wheels.
As far as chains go, however, in nearly thirty years of living in the shadow of the Alps, and driving regularly in snowy and icy conditions, I’ve never had occasion to use chains, although they are compulsory during the winter on certain, usually very steep, back roads.