The offending patch had something to do with Microsoft.NET. Every so often, I would dutifully try to install the update, only to have it fail for no reason at all (messages like "The update failed to install" were not very helpful in diagnosing the problem).
I dug around a bit, figured out the hotfix number, and tried downloading the patch .exe directly from microsoft.com and installing it, only to have it prompt for an .msi file halfway through the install. Again, no help was given on how to proceed or fix the problem.
I was finally able to track down some help with Google (link1, link2): I had to download dotnetfx.exe, unzip it to a directory, download the patch, run the patch, and (when prompted) point the patch to the unzipped directory to get a file that it needed. Once I did all of this, the patch finished installing and I could return to the serenity of a taskbar tray free of the yellow shield.
My big question is: What gives with the error reporting? Why wasn't more information given on the reason for the failure and how to fix it. Even better, since my copy of WinXP knows that it was in trouble, and since I am a registered user of a valid copy of XP, how come my OS couldn't "phone home" and report the problem so that somebody from Microsoft could help me fix the problem.
Anyway, I'm posting this here in the hopes of helping out any other poor souls who are currently trapped in a similar situation.