Originally Posted by HTupolev

A large contact patch doesn't increase friction for two reasons:
1-Ideally, the rubber of the contact patch isn't actually sliding against the ground, it's sitting statically on the ground.
2-Even if it was sliding on the ground, friction tends to increase with larger total forces, not larger contact regions.

Wider tires deform in a shorter, rounder, smoother region at the ground contact; as a result, less energy is wasted deforming the tire walls as the tire rolls and as it deforms from bumps.

Also, as you get toward bumpier and bumpier surfaces, lower PSI is important for low rolling resistance. Much better to absorb the bumps with tire deformation than let them transmit higher up, jostling the bike and rider around! And wide tires are able to handle lower PSI.