How do I fix a malfunction in the Windows 11 shutdown update feature?

Seen this exact thing before, the shutdown button and the update page both rely on the same Windows Update service chain, so when it breaks they usually go together.

First thing I'd try is bypassing the Start Menu entirely. Open PowerShell as admin and just run shutdown /s /t 0  if the button's the issue, that'll get you a clean shutdown without touching anything else.

If that works but the update problem's still there, the most likely culprit is a partially-applied update that got interrupted and left things in a weird state. To clear it out, paste this into admin PowerShell:

net stop wuauserv & net stop bits & net stop cryptSvc

ren C:\Windows\SoftwareDistribution SoftwareDistribution.old 

net start wuauserv & net start bits & net start cryptSvc

That wipes the update cache and forces Windows to rebuild it fresh. Then run DISM /Online /Cleanup-Image /RestoreHealth followed by sfc /scannow, do them in that order, DISM first.