On screen: An error occurred while updating [game] (missing file privileges)
This error is produced locally. It does not appear because the service is down, so the fixes below are worth working through in order.
This is a local permissions problem and has no relationship to Steam's service status.
Steam does not have permission to write into the library folder, or a file in it is locked by another process. It is a local permissions problem rather than anything to do with Valve's servers.
Common causes: NTFS permissions on the library folder, a read-only attribute, antivirus or Windows Controlled Folder Access holding the file, or a library on a drive that was moved between machines.
Install the same game to a different drive or a newly created library folder. If that works, the permissions on the original folder are the problem. If it fails everywhere, it is your antivirus or the Steam service itself rather than any one folder.
Operator documentation: help.steampowered.com. Valve does not document this error specifically. Their only relevant article covers update and installation issues generally, and says: the error results when Steam is not able to complete the installation, and the specific reason will be shown in the parenthesis at the end of the error message. They confirm the format and never explain the reasons. So anyone quoting an official Valve definition for the text in those brackets is quoting something that does not exist. Valve does separately publish the Repair Folder tool and a list of programs that interfere with Steam, both of which apply here.
Steam does not have permission to write into the library folder, or a file in it is locked by another process. It is a local permissions problem rather than anything to do with Valve's servers.
This error is produced locally. It does not appear because the service is down, so the fixes below are worth working through in order.
Start with: Use Valve's Repair Library Folder. Steam, Settings, Storage, select the drive, the three-dot menu, then Repair Folder. This is Valve's own tool for exactly this failure and it fixes the permissions case directly.