I live at sea level and there is a large mountain range in my back yard. Going from wet/dry to snowy/icy and back even several times a day is normal. Here are the things I have learned.
1. A decent/cheap set of all season tires combined with a cheap set of studded winter tires will beat the year-round performance of any single tire. Period.
2. If you wish to use a single tire year-round, a more winter-leaning tire is going to be much worse in warm rain and hot dry pavement than a more balanced all-season with a set of 4 chains.
3. The General Altimax RT43 often tops the results in Tirerack tests of all-seasons in snow and ice.
https://www.tirerack.com/tires/tests/testDisplay.jsp?ttid=215
4.
Buy the skinniest tire allowed for your vehicle!!!!! This is a HUGE aspect of making an all-season better in the snow and ice. Wider tires are usually only helpful for deep snow when ground clearance is running out. Otherwise skinny tires will almost always grip better than wider tires of the same brand and model. This is especially critical for front wheel drive and AWD with a front bias since you can't add weight to the front of the vehicle.
5. Purchase a nice set of tire chains (4 of them and not cables) and a nice set of chain tensioners. The tensioners are critical for making the chains work better at higher speeds.
6. If not using dedicated snow tires, I prefer to buy the best all-around tire that doesn't suck in warm/dry or warm/rainy and then carry a high-grade set of chains and an air pressure gauge along with air pump. Air down the tires in snow or ice then toss on the chains when airing down isn't enough.
7. All-season tires are often good enough for the first winter, but once that tread wears down all year and into the next winter, they are going to be far inferior depending on the miles driven. Most all-season tires will last way beyond their ability to handle snow at all.