Well yes, weather will cause rust. Good use of brakes can clean it up and keep the mechanism from seizing. Lack of braking = lack of use as far as the brakes are concerned.
I could run my errands without braking at all (except when stopping at a parking lot). Anticipate lights and traffic flow, downshift to slow down.. that's how I used to drive. At home I could even park without braking to slow down, thanks to a small incline at my parking spot.
My previous car, which had automatic transmission, didn't have really show any issues with rust because I actually had to brake when driving around town. Same weather, same locale.
Rusting is for me more associated with the weather and lack of use of the whole car than with how much do you use breaks when you use the car.