Thanks, Jake, that was the boost of insight I needed to get it right. I had drilled out the bottom of the rivet with a bit that was smaller than the inner diameter of the rivet. Instead, I needed to drill down with a bit that was slightly smaller than the diameter of the *hole* to drill out most of the side wall. Aha!
Of course, I didn't know how big the hole was, but just kept scaling up the size of the bit until the rivet began to spin easily.
I tapped the head off (sideways) with a screwdriver and then tapped what was left of the rivet down into the boom. Voila.
Thanks again, Jake. I owe you a beer.