I rode an old English 3 speed all winter in Madison WI for many years. Upright position was probably the most important factor. Different tires are not going to make that much difference. Riding in fresh snow, packed snow, slush, ice is really about assessing traction and then using the right bike handling skills to get through it. Stay in the saddle, don't think you can lean going into corners and easy on the front brake.
Riding through 2-4 inches of fresh snow is really easy. Good traction. Glare ice or really hard packed snow is the worst.
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Il faut de l'audace, encore de l'audace, toujours de l'audace
1980 3Rensho-- 1975 Raleigh Sprite 3spd
1990s Raleigh M20 MTB--2007 Windsor Hour (track)
1988 Ducati 750 F1