If it were mine I would want to diagnose what might be sticky. If you loosen the nut holding the brake cable at the caliper you can see if the cable slides easily in the housing and if the caliper arms move freely. If either is sticking I guess I might try a drop of oil on the calipers at the pivot point where the bolt holds them together. The type of oil isn't as important here, maybe even WD40. If the cable is sticking in the housing I might pull it out of the housing and check it for any rust, and replace the cable if needed. It looks so clean though so I can't imagine it would be too bad.
At the hub you could pull on the little chain that goes into it with your hand to check for freedom of movement there. Sometimes you'll have to pedal a little to get the hub in the ready-to-shift sweet spot. A few drops of oil into the port on the shell may help. There is a 20 weight 3-in-1 oil with a blue label that is good for these hubs, just don't use the regular 3-in-1 as it can gunk up after sitting a while.