It appears that there was some truth to both my hypothesis and the alternative: my caffeine addiction was likely causing withdrawal, but the caffeine dose seems to have been preventing idiopathic headaches as well.
I got to day 37 of my caffeine weaning adventure, which means I got down to 7.5% of my normal caffeine intake before I ran into any trouble. This mostly had to do with a few weekend trips that made accurate measuring of the instant coffee powder problematic. Anyway, after a few headaches and an extra week of slow weaning, I got completely off of the instant stuff.
So I’m still having a cup of decaf nearly every day, and on those days that I don’t have the decaf I mostly don’t have trouble. Seemingly independent of the decaf, however, I am randomly getting minor headaches a couple times a week. Usually starting in the early afternoon. So far I’ve been able to knock them out with a few hundred milligrams of acetaminophen.
In conclusion: it is likely that some low level of caffeine is somehow preventing some real source of headaches, and that I had also developed a dependency on the extra caffeine that I had been consuming which led to withdrawal headaches. I should probably figure out the biological source of the headaches that remain…