I'm pretty sure that it's the tire being worn. There are a number of spots where the threads are exposed that seem to line up pretty well with the punctures. They're not sharp, but I can envision a situation where the roughness caused by the exposed threads is grabbing onto the tube and causing a flat. I've already ordered a new tire, but just to be sure, I'm going to try covering the exposed threads with patches and seeing if that does the trick.