The friction model you have been taught in school is the Columb model which is a fairly good approximation of how friction works for most cases. And in this model the friction does not depend on contact area. However there are some assumptions in the basis for this model which does not always stay true. For example in motor racing it is easy to saturate a small contact patch if you apply too much force to it. The materials will then bend and deform from the huge force alone. In fact this is often the case when people drive on the edge of friction and this can cause rubber and pebbles to be thrown free from the tyre and track. Increasing the contact patch means that there is less forces going through each section of the tyre and track meaning it can takes more forces before it gets damaged.
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