I think both sides of the argument are correct here, you just don't realize that you're arguing two different points.
True, friction is simply the "normal" force times the coefficient of friction. (Normal force meaning the force perpendicular to the contact surface - in this case the normal force is the gravity times mass). And true, it takes the same amount of force to slide a box on its wide side as it does to slide it on its narrow side, since friction has nothing to do with surface area.
However, this does not stand up very well when you get to deformable objects like a tire. A road surface, no matter how smooth, is actually made up of a lot of peaks, valleys, points and divots. A tire is able to deflect and confom to these irregularities, whereas a box simply rides on top of them. To accerate the car then, the tire is not relying only on the friction force between the tire and the road, but also on the fact that it is able to mesh with these irregularities, almost like two gears meshing, and "push off" of the irregularities helping the car to move forward. A wider tire has more irregularities to "push off" of, and thus can provide more traction to accelerate the car either in a straight line, or while cornering. Granted, most of the acceleratory (is that a real word?) force is from friction, but much of it comes from the tire pushing against the irregularities.
At least that's my theory on it.
And from the "Ask A Scientist" and the explanation that merely increasing the contact area will increase the force required to slip, this holds up for the tires if you apply my above explanation to the problem, it does not hold up for the box, whether you believe my explanation or not. The only thing that governs when the box will start slipping is if the force pushing the box exceeds the force of gravity multiplied by the mass of the box multiplied by the coefficient of fricton.
The only thing in this thread that I think is just wrong is arguing that top fuel racers use wider tires because they are heavier, which in turn gives better traction. If a drag racer wanted to increase traction by adding weight, he would simply add some weight over the driven axle, not use heavier tires. A heavier tire would have a have a higher rotational inertia than a lighter tire, making it more difficult to accelerate. Not to mention that adding weight to a dragster is a ridiculous idea.
Bob