How did this repair eventually work out ??
Sorry, forgot to come back and update.
After acquiring the Weller WES-51 soldering station and some supplies, I went through and replaced all of the questionable KZG capacitors, including the three 820 uF ones that had visibly failed. I used 63/37 multicore solder, and a solder sucker to pull the excess solder from the through holes. I replaced the 820 uF caps with Nichicon UHM0J821MPD.
On a couple of the caps I had to work on them a while to get them to pull through the board, but not terribly bad. I added flux and fresh solder before pulling and that helped a lot.
Computer runs like a champ. Problem is my wife got used to my Core i5 laptop serving as her desktop computer, so we will probably end up upgrading in the next year anyway. Still, nice to have it functioning until we have a plan to replace it.