I watched this video yesterday and was absolutely steaming frustrated by the end of it. This video seems to me like a ploy to sell more expensive winter tires in cool looking sizes to owners of sport crossovers and sport sedans by creating a test that will make them all appear comparable. There are several flaws with the test that need to be considered before drawing conclusions...
The narrow tire in this test, is both not the same tread pattern and is a smaller diameter than the other tires tested. Since the tire is a smaller diameter, it comes to the table with no additional load capacity per inflation pressure compared to the larger diameter, wider tires it is competing with, so it must be inflated to the same pressure as the other tires (the 205 and 255 in this test both have the same load index ratings). Furthermore, the prepared surface conditions both of their track and ice do not reveal any of the effects of changing the size or shape of the contact patch, as those conditions are designed to compare tread patterns and rubber compounds, not contact patch shape. The real world does not have a groomed track with a very consistent amount of snow over a hard surface or a very specific type of ice on it.
The "skinny is better" is not a good way to conceptualize it as that's not the primary focus of changing to the "skinny" tire. In reality, it is wide low profile tires on reinforced sidewalls and high inflation pressures whose contact patches are very skinny and rectangular in the wrong direction. The goal is to get to a 60-75+ series with more load capacity so it can safely be used at lower pressures. 60-75 series passenger car tires operated at appropriate pressures for the load produce uniform contact patches that are roughly as long as they are wide (close to square). This provides the MAXIMUM floatation effect. I'm sick of all this nonsense about skinny tires "cutting" though. That's not the effect that makes them superior. The skinny tire has to be taller profile and operating at lower pressure to actually gain a meaningful advantage. A 60-75 series tire operated at appropriate pressures gradually compacts and climbs on top of the snow/slush in front of it. The deformation and larger contact patch of the "skinny" (SQUARE!) tire paves its way OVER unplowed or heavy/loose snow/slush situation, while the rigid, nonflexible low profile reinforced sidewall XL tires at high inflation pressure have almost no useful contact patch longitudinally. The wide rigid tires act like a wedge driving down into the snow digging their own grave. Watching an AWD crossover on low profile tires with the traction control unable to figure out how to lift its tires out of all the silly little crescent shaped holes it has dug itself into is hilarious.. I've seen crossovers with low profile tires totally stuck on a mild incline with a few inches of wet snow. Happens all the time because the tire is not paving/packing/climbing and the traction control is only reactionary, so it not only ALLOWS every tire to dig itself a grave before reacting, it has no other way to work. (Hey, if you've read this far remember that next time you're trying to drive a car with traction control or ABS... Proactive traction is always better than trying to find traction reactively that has already been lost!).
If you always drive on prepared track like winter conditions (frequent plowing is common in some areas, not here!), then low profile tires will work fine as long as they have a good winter rubber compound and tread pattern. If you deal with unplowed roads on a regular basis (common for residencial and rural roads, and even roads in major cities during a snow event), a larger SQUARE contact patch is what you want, and the way to get that is a narrower tire on a smaller wheel and a taller profile, operating at lower pressure if it can be done safely.
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I run 315/75R16's on ice and snow all the time. I can "pave" a path through snow significantly deeper than the available clearance of my SUV using this tire setup at ultra-low pressures. I will go as low as 8PSI to get the contact patch length maximized for busting through 2-3' of snow. For more "normal" conditions when its icey/snowy I'll run 15-25PSI and keep speeds appropriate for the inflation pressure, but this is still coming up short of an ideal square contact patch...
Best tire size I have ever run for driving mountain passes in heavy snow of 6-12" was a 235/85R16. This is not because they are "cutting through" anything. This is because at this size, under the weight of a typical SUV or light truck, this size will produce a very near square contact patch at around 30PSI, which is the sort of pressure that a tire inflated to 35PSI at more normal temps, will drop to during a winter weather situation.. There's no need to air down to get a uniform contact patch that maximizes floatation with this size tire. It's already operating at an ideal contact patch shape for winter weather at normal inflation pressures.