At least one problem was obvious: the plastic gears were crumbling (this being a common failure mode).
Fortunately, I had on hand set of Tom Kelly's carousel repair kits. These kits are well done, but warn that the repairs aren't particularly easy and run a real risk of scraped knuckles and frustration. Unsurprisingly, I decided to go for it.
The first step was to remove the surrounding sheet metal and the focus motor bracket. The screws were tricky to get to, so I ended removing the bottom support bracket and tilt adjustment wheel, which made everything much more accessible.
The next step was to remove the old spur gear. The instructions suggested cracking it off with wire cutters. As it happens, it was so fragile that a quick tap with the pliers shattered it, and only a little effort was required to remove the remaining plastic from the shaft.
To install the new gear, it needs to be pressed onto the shaft using an adjustable clamp and the supplied support block.
I was a little worried about this procedure, but it actually went surprisingly easily. The next steps were to remove the old worm gear (a similarly easy process, as the plastic had almost no integrity any more) and (bracing the motor shaft) tap the new one into place. This also went smoothly.
Testing confirmed that the focus function now works correctly and quietly. Success!