Disc brake rotor size

I just noticed that Avid's mechanical disc brakes come with 160, 185
or 203 mm rotors. It's claimed that the 185mm rotor has 15% more
"power", the 203 26% more.

Does a larger rotor really provide more stopping power? Why? I could
understand less fade due to heat, since a larger rotor would dissipate
heat better. Does that translate into more "power"?

I assume that dimension is the rotor diameter. But wouldn't a change
in rotor diameter change where the caliper gets positioned, i.e. would
require a change to the caliper mounting eyelets?

I'm spec'ing these brakes for a road bike; should I expect to need
more stopping power than a mountain bike for say long mountain
downhills?

TIA