Snow Tire Wraps for Bicycles
You can avoid this problem altogether with studded tires



The pattern is deep enough so that it grabs quite well on the snow, and the studs give you a very good grip on ice and hard snow. On uniformly flat ice, the grip is almost as good as asphalt.

If the tires are good, you can wear them for all winter season, even if there is no ice. The studs are not very prominent, so the whole tire is still in contact with the road. This solves the problem of having to adapt to weather conditions. At my local bike store they assured me they would last for at least 5 years.

Source: I have tires like the one in the image in the Swedish winter.

PS: here, I have never seen cars them using chains or socks, they have winter tires instead, pretty much a car version of the image. Not all have studs, though.