Downsizing to 20 inch wheels for your car will not help traction. The wheel diameter itself doesn’t matter at all. What matters is the tire size, not what wheel it is mounted on.
Looking at Tire Rack, the winter tire selection for your car in 21s versus 20s, the width and aspect ratio, which do matter, barely change. The width doesn’t change at all and aspect ratio changes by 5 points. Basically, no change.
So buy your winter tires on 21’s and have at it.