Snow tire buying guides often recommend that drivers purchase rims that are 1-2" smaller in diameter than their car’s OEM rims, and then purchase snow tires to fit those rims (with the tire profile chosen so as to match the OEM tread diameter).
The claim is that doing so enhances snow traction, but I can’t find an explanation as to why.
I don’t think I’ve ever seen a recommendation for a smaller diameter wheel, although sometimes recommended snow tires are narrower than all-weather or summer tires.
It might stem from the decreasing section height of tires on many cars; you don’t really want 40-series snow tires for a number of reasons. Going to a smaller wheel and a taller tire section provides more cushioning, especially for wheels that might more frequently strike curbs, ice dams, debris etc. in snow conditions.
A narrower tire will tend to “cut through” snow and slush for better traction; if you have a vehicle with wide summer tires that you want to drive in snow, significantly narrower tires may be precisely what’s called for. I suspect that large-diameter, low-profile, wide snows are either unavailable or very expensive.
I have a Volvo XC90 that came with 18-inch wheels. When it came time to look for a set of snows, I found the 18-and-up sizes were more limited in availability and higher in price than 17-and-down. I found a set of 17 inch OEM wheels, bought Blizzaks to fit them, and got away for $500-1000 less than any 18-inch options I could find. (I wanted mounted snows, not tires that had to be swapped.)
So: section height, narrower tread and section, and cost all go into using a smaller wheel diameter for snows.
Assuming everything else is more or less equal, tires with the same outer circumference on smaller rims will have more flex in the sidewalls, which will offer more ability to cope with lumps and bumps like chunks of ice or ruts, and may have a larger contact patch with the road.
I understand narrower wheels, but smaller wheels wouldn’t seem to do much. You don’t actually want sidewall flex like you do in an offroad situation, because widening the contact patch in snow decreases traction. Usually snow tires are inflated a few psi more than their all season counterparts for that reason.
Tire rack does have some guidance for buying smaller diameter wheels, but note that in their example, they’re going from a 225/45R17 to a 195/65R15. Smaller diameter, yes, but also 30mm narrower, which is the key point. It may just be that wheels tend to scale in both directions – most OEM 17" wheels are in the 6.5 to 8" wide range, whereas most 15" wheels are in the 5.5 to 6.5" range. So just by virtue of what’s available in the market, narrower often means you also have to go smaller.
I think you’re assuming too much similarity between summer and snow driving. Most snow driving is closer to off-road than not - done slower, with less need for absolute cornering ability, and over bumpier and less stable terrain. The small losses in traction and stability from another inch or so of tire height vanish in the special needs of staying stable in a snow and ice environment. The cushioning improves both ride and maintenance of lower-speed traction, the higher tire height puts the wheel up out of easy damage range, and significantly narrower tires cut through unstable upper layers of snow rather than sliding over them. You want a somewhat higher weight (per square inch) on your contact patches in snow, slush, and muck over a hard surface. The loss of traction from reduced contact area becomes irrelevant, next to the other considerations.