Based on inputs from Cash and Daveh I got on this forum, I adjusted my rear brake by loosening the spindle nut, putting full pressure on the brake lever and then tightening the nut. This is meant to center the shoes in the drum.

It worked! For the first time ever I can actually feel some braking force coming from the rear brake. I suppose I may be able to improve it some more by enlarging the spindle hole and repeating the process, but I'm not going to mess with it. I mostly use just the front brake anyway, but now I can use the rear to balance the braking if I need to. It's not such a bad thing to not have a hugely powerful rear brake anyway.