Cool! I get to do the engineering nerd thing.
The thing that initially appears counter intuitive is that friction force is independent of the contact area.
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It is directly proportional to the normal force. The normal force is related to the ratio of piston area between the caliper and the master cylinder.
Brakes with larger pads are simply a result of them having more piston area. If you designed a 6 piston caliper but kept the pad size small, you would have a lot of trouble achieving even wear and your pistons would likely cock sideways and jam as a result. As a bonus more pad material lasts longer because, well, there's more of it. For a given rotor diameter, a multi piston caliper is better than a single piston of the same pad area because the geometric centre the pad contacts at a greater radius. Or put another way, the middle of the long narrow pad is closer to the OD of the rotor than the middle of a square pad.
Now for the rotors.
There's a few things we want/need from our rotors. We want them to be of a large diameter and fully support the pad to keep the pistons nice and square to the caliper. We want them to not overheat or buckle, so make them nice and fat with no holes in them, that gives us lot's of thermal capacity and a strong brake rotor/flywheel. But wait! We also want them light, so make them thin and put lots of holes in them, but also make sure those holes don't clog up with mud or compromise the support of the caliper pistons. The holes also aid cooling by increasing surface area, but you are trading off thermal mass.
So it's a balancing act between performance, weight and cost. (I'm lumping robustness in with performance here)
@Freediver is on the money! Clamping force is what gives you the friction force. F = P*A For a given force, increasing pad size just decreases the pressure.
Sublimation of the pad material likely only occurs under very specific conditions. There are a hell of a lot of vehicle brakes that work just fine with solid rotors so I suspect this is more marketing BS than any real significant factor.