Well... some good news and some bad news... I made a poor decision and learned a lesson about trying to re-flow very small pins gently with a soldering iron. I will not be fixing that IIsi any time soon, unfortunately.
On the other hand, I re-capped the other IIsi, which was the one in worse shape of the two as far as cap goo goes, flawlessly and it's up and running.
I appreciate everyone's help with troubleshooting the first one anyway. I'll worry about getting back to it when I'm a bit more experience with dealing with smaller pinned surface mounts.
I've got a IIsi that soft powers on, speaker crackles a bit, and it shows a light grey screen for a moment before the screen turns black. Is that the same symptom that you experienced, @shadedream?