Well, of course, you learned in physics class that the force of friction = frictional constant times the normal force. Thus, surface area has nothing to do with friction. Double the surface area, and you have half the pounds per square inch. I have my students do a lab and I do a few demonstrations in front of them to prove this.
However, when you're talking about rubber tires on the road, it's not as simple as saying "it's only friction." Traction is a bit more involved. If it wasn't, tire companies wouldn't be hiring engineers to design better tread patterns, as the tread patterns would be irrelevant. They'd just make pretty designs and claim they work.
It's a pita to explain within 1000 words, but I'll try:
Traction isn't just a synonym for friction. Friction makes up part of it, but not all of it.
I think a good analogy that can be understood is this:
Imagine a really slippery linoleum floor. Now, imagine a linoleum floor that's just as slippery, but is dented all over the place with little quarter inch deep depressions.
Lay down on that floor and pull yourself across the floor, but keep your hands perfectly flat. That's friction. Now, pull yourself across the floor, but allow your fingers to bend to try to get a hold on those depressions. That's traction. If you allow your hands to change their shape to fit into the irregularities, you can "grip" the floor.
On a road, over a wider area, there are going to be more of those spots to grab by the rubber tires. However, if it's over too wide an area, there's not enough pressure on the rubber to get it to thoroughly dig in. Too hard of rubber and the tires will take a lot longer to wear out, but they won't "dig in" as much. Too soft of tires, and they'll give you much better traction, but will wear out a lot quicker due to more shearing.
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Here's another mental exercise to show why you can't just say "surface area doesn't matter":
forget about the rubber tires for a moment. Imagine a road covered in packed snow. You have 4 tires on the car. Put chains on the two drive tires and see how much they "grab" the snow to help push (rear wheel drive or pull - front wheel drive) the car forward. Now, somehow extend your axle to allow for 4 drive tires. Put chains on all four of the drive tires, essentially doubling the surface area of chain/snow. Think you're going to get more "grip" i.e. traction? You betcha!