The supercap does go bad from time to time. I just replaced mine since I was replacing everything else.
For the OP, without knowing your soldering skills or your troubleshooting skills, be aware that duos are prone to have trace damage by now, so expect that you will be spending extra time/effort to fix that. Additionally many people are switching from the through hole style caps to using ceramic or tantalum. These are surface mount but will not need to be replaced again. They are harder to solder than the through hole style. If you want a set, I have a full ceramic kit I am selling.
As for the power brick, many of us switch these out. Technically you should be fine to use the original, but honestly I hate those giant transformers so I switched to a slim ps2 adapter for my duo r.