I can now confirm that all three of the iMac G3s are alive and well, but the third's revival was a bit more dramatic than the others, although it may perhaps point to the actual problem with these things (to someone more knowledgeable than myself).
It took almost a week to start up stably. When it started turning on, at first all seemed well, but then there were snaps/pops and momentary distortions in the CRT raster. I was worried something was shorting, but I never caught it visually inside and it eventually stopped. (At this point it was not reliably staying on.) However, the display would then begin to sink and distort on the screen, with horizontal worms across the whole display, lasting generally until the machine was shut off again. This too has disappeared, and everything appears to now be completely stable. I backed up the HDD, installed the original system, and updated through 9.2.2.
Before I update the firmware to 4.1.9f, is there reason to try and preserve the factory boot ROM? It's version 1.2f2, though I would have no idea how to do this.