I did a repair installation of XP.
One note: if you have Internet Explorer 7 or 8, you'll need to either uninstall it before you do the repair installation, or reinstall it after.
It seems to have been a problem with WMI, as System Information was unable to display any system information. One repair install later, and System Information and CCleaner are doing what they should.
Recently, CCleaner has not been displaying any restore points on my XP SP3 machine, despite Windows's own System Restore tool showing 4 restore points.
Turning System Restore off and back on has not helped, nor has uninstalling and reinstalling CCleaner.