Theoretically speaking, area doesn't affect friction force. P=F/A so F=PA. Friction Force = (PA)x coef.fric. As you can see area will cancel out. The only problem is in snow, the wider the tire with more grooves, it makes it easier for snow to stick on to the tire and reduce the coefficient of friction. Skinny tires are better at displacing the snow away to the sides as it rolls.
That is true, theoretically speaking.
The amount of frictional force is proportional to the contact surface area, but it's also proportional to the pressure with which the two surfaces (the tire and the road) are pressed together. If you increase the surface area, you'll (by definition) decrease the pressure since pressure is the weight divided by the surface area. So yes, a change in the surface area will cause two counterbalancing effects. That then leaves only the weight of the object to be considered.