They don't cover how a tire works in physics class because it is a specialized topic. A sticky tire works kind of like velcro, especially on a concrete track. In physics class they cover the "classic" friction situation of hard surfaces sliding over each other. With velcro, the more surface area you have then the more friction you have. Tires are not exactly velcro, but it is one way to think about the issue.

Tire design is actually a very complex subject. That is why my advice to the OP was to call the tire company tech line and ask for advice.