True.
> which in turn means a fairly high-capacity battery.
Not true.
A microcontroller CPU is good enough. Take a look at the KK2.1 flight controller.
https://www.hobbyking.com/hobbyking/store/__49254__Hobbyking...
Its brain is an ATmega644, which only requires 240µA @ 1.8V, 1MHz, according to the spec sheet.
http://www.atmel.com/devices/atmega644.aspx
You do need energy-dense batteries, but all that energy goes into the rotors.