Well, I have done a little more research. I used DVDInfoPro to determine the exact Manufacturer ID of the Verbatim DVD+R's I was using, and then looked that up
here. I was surprised to learn that most people seem to really like Verbatim discs, and the vast majority of people found them to be very reliable. The few that reported problems, though, reported problems with the "outer edge" of the disc, exactly as I have.
A few people mentioned that if they burned at 1x or 2x instead of 4x, the discs played flawlessly. I hadn't even considered the possibility that burn speed could make a difference (I have to keep reminding myself that these are not CD-R's), but I'm burning a disc at 1x right now, just to see. Burning a disc in an hour seems a lot less convenient than burning one in 15 minutes, but if it fixes my problem, I will be happy nonetheless.