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Strictly speaking, Joe didn't really say anything about needing "bigger" brakes, just more fade resistance.
As for the swept area comment, I disagree, to a point.
As you say, pad area helps with fade resistance (or can, anyway). By itself, however, pad area does nothing to improve braking torque. Brakes follow the "classic" friction model, in which area of contact is NOT a factor. Only force of contact and the coefficient of friction between the two materials (pad and rotor) matters. If you pressed a pencil point made of brake pad material against a rotor with the same force, you'd get the same amount of braking torque, as you would with a huge pad. The difference is the pencil point would wear very rapidly. I've seen some racers attempting to solve a brake balance problem by milling away some amount of pad material to "reduce the braking". All they end up doing is wearing out those pads faster, and often causing fade by heating up the remaining material too fast. This does end up reducing the braking, but is obviously not the ideal way to do this.
Now, there are also leverage effects at play here. If you apply the force farther from the axis of rotation, you get more braking torque for the same amount of braking force. Look at any recent sportbike front brakes and you see an excellent application of this idea: a thin, annular ring of brake rotor well away from the hub, held there with a spidery "hat". This ring requires a long, thin pad to get adequate (for wear and fade resistance) area, so you need a long, thin multi-piston caliper to squeeze that pad, or you need a thick, rigid (and heavy) backplate for the pad to spread the load for you.
Joe's problem is just one of shedding heat, so there's no need for bigger calipers and/or bigger caliper pistons. Bigger pads would help some, ducting air will likely help a lot, vented rotors would help a great deal.
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