I came across this question when considering new vehicle tires in a snowy environment. It appears that big off-road tires have very deep treads which greatly increase the surface area of the tire. Does this allow the tire to have more traction since more surface area is in contact with the ground resulting in the concept of "more surface area equals more traction"?
No. Tread is there to account for the presence of "other material" between the tire and the road surface. For example, if there is water on the road, a perfectly smooth tire would be susceptible to aquaplaning: the water has nowhere to go, and becomes a thin film between the tire and the road.
If anything, deep profile tends to reduce the area of the tire in contact with the road, and increases the pressure. Both of these things give the "other stuff" (water, snow, loose gravel etc) a chance to get away from between the tire and the road, allowing the tire itself to grip the road.
In situations where there is a thin layer of snow (and to a lesser extent, mud or clay), a deep tire grip can shape the snow into a waffle like pattern that interlocks with the tires, and results in (much) greater lateral force - no "simple friction", but the kind of resistance to relative motion that comes about from having interlocking structures (like the teeth of gears). The deeper and wider the profile, the better the chance of a solid ridge of snow/mud being built up.
\Once you have the "intended" interface between tire and road (without the other material), the force of friction is largely independent of area, since the total normal force will be constant. But most of the time, tires slip because you are not dealing with just the road; and the profile is intended to deal with "the other stuff".