Originally Posted by Red__Zed

People always say things like this because the Physics 101 equations (F.friction = mu * F.normal) say friction isn't dependent on surface area, but the Coulomb model has a couple of holes in practice, especially in regards to car tires.

Increasing contact patch does help with a tire for a couple of reasons, especially because we are concerned with traction, rather than just friction.

One element is maintaining stability over rough surfaces. A wider tire gives you more opportunity to have rubber in contact with the road, as the tire deforms over imperfections. If you've ever cornered on a road bicycle and hit something mid corner, you'll understand why this is important.

Adhesion plays a role in traction as well. If you slap a piece of tape on your desk and try to slide it, you can see this. It will be very difficult to slide, despite a very minimal normal force acting on it. Adhesion comes into play on car tires as well - there are dozens of SAE papers out there on it, and the Wikipedia page on friction has some links for you as well.

If you really want to get into the weeds on cornering performance, a wider tire also tends to help with slip angle. And of course heat management comes into play.

Other reading:
https://en.wikipedia.org/wiki/Tire_load_sensitivity