I tend to agree with the guy. If a given contact force is spread out over a greater surface area, the psi goes down, the total force remains the same.
Friction is pretty much dependent on coefficient of friction of the materials & the clamping pressure & independent of friction surface area.
A larger diameter disc or drum does give the brake more mechanical leverage on the wheel & would require less friction, clamping pressure, to get the same stopping power. A smaller disc diameter would require more pressure. Also a larger swept area gives more cooling & less wear.
In case you are wondering, this does not apply to tire grip. Soft tires actually have a good measure of adhesion in addition to friction, & can benefit from more surface area in contact with the road. Wet or icy surfaces, where there is no adhesion, excepted.
D