Meanwhile my understanding of Non-Steroidal Anti-Inflammatory Drugs (NSAIDS) was that they were exactly that by definition - Anti-Inflammatories with pain being reduced by reducing the immediate cause of it, inflammation, not by interfering with the actual mechanisms of pain.
I guess one could consider general anaesthesia (laughing gas, ether, and more modern successors) to be "painkillers" too but they're obviously not quite the same category of thing.
Anyway, I enjoyed the article - it would be interesting to hear the author give their own reasons for not accounting for NSAIDs.