All of the talk of "small" tires is in reference to a winter setup. You want to run as narrow a tire as you can with winter tires. This yields more traction in snow because the tire cuts through the snow and onto the road. A wide tire will be more likely to skate or float across the top. So in this case, small does not equal low traction.
Quality, of course, still matters. But just about all of the winter tire manufacturers make good products, it's hard to find a "crappy" winter tire. Some are just better, and sometimes more expensive, than others. In any case, a cheap winter tire will outperform an expensive all-season in the snow, guaranteed.
Mine: 2006 Mazda3 s GT Hatch: 115k, Gray, Manual
Gone: 2002 SVT, #1654
Hers: 2012 Focus Ti Sedan: 55k, SGM, Manual