perform this thought experiment: you have a very tall tire and a very squat wide tire. You test both tires on perfect, slightly wet ice. No traction. Now, use the same tires on a dry asphalt surface. The surface is comparatively rough and provides millions of hookup points for tire treads.

At the molecular, nano and micron scale there are no irregularities in the surface of the ice into which the molecular, nano and microscopic irregularities on a tire's outside surface can interlock and push against. It is the interlocking of these irregularities that we call friction and provides the transient instantaneous pushing that converts the rotational tire forces into the linear force that creates forward motion. This is why the wider contact patch works - it provides a larger statistical opportunity for millions of micro hookups that a taller tire cannot (practically) provide with its smaller contact patch.

I have often wondered why drag racers don't experiment with sticking (super strong) sandpaper to the track (grit facing up) for the first 30-60ft of a run. Has anyone ever heard of someone trying this just for fun?