Windows (Vista Basic SP2) Update Fixit reported that it had diagnosed and repaired all Windows Update issues, but Windows Update still could not check for updates because the Windows Update service supposedly wasn't running, but it was running according
to the Services administrative tool, as was the Background Intelligent Transfer Service (BITS).
So we set the Windows Update control panel to never check for updates, after which Windows Update reported that it needed to download and reinstall Windows Update Agent, which it did, and Windows Update worked again.
Could the Windows Update Fixit be upgraded to diagnose and repair what we somehow fixed via the Windows Update control panel?