OK, believe it or not, the smoky smell just went away by itself!
I took "Smelly" outside, took off the case, turned it on, and let it sit for 20 minutes. There was no bad smell, other than a faint unpleasant smell if I put my nose right up to the Mac. I also tried swapping in a different logic board, to see if the smell was coming from Smelly's analog board or logic board. But since everything worked fine and I never smelled more smoke, the test was inconclusive.
The only suspicious capacitor I noticed on the analog board was C38, right behind the power plug. It's got some ancient-looking package that I've never seen before, and it looks cracked.
I then took Smelly back inside the house, put it all back together, and it's been running for about 90 minutes so far. As long as it continues to run and doesn't smell like it's about to burn down the house, I'm not going to attempt any pre-emptive capacitor replacements.
By the way, what do people recommend for cleaning corrosion and battery acid off terminals in the battery compartment? A wire brush, or some kind of solvent maybe?