`--log` Command Does Not Save Logs

Brief description of your issue

Observe the following command:

winGet update --id 7zip.7zip -h --accept-package-agreements --accept-source-agreements --include-unknown --disable-interactivity --log "$env:PUBLIC\log.txt"

The documentation states that the --log option applies for both install and update commands, with the following text:

Directs the logging to a log file. You must provide a path to a file that you have the write rights to.

The command I posted does not create a log file. Why not?
In StdOut I see the following:

No available upgrade found.
No newer package versions are available from the configured sources.

But I expect to see a log file created which contains at least this information, if not the error code as well.

Steps to reproduce

Run this command:

winGet update --id 7zip.7zip -h --accept-package-agreements --accept-source-agreements --include-unknown --disable-interactivity -o "$env:PUBLIC\log.txt"

Notice how it does not make a log file in C:\Users\Public\log.txt.

Expected behaviour

A log file to be created with logging information

Actual behaviour

No log file is created

Environment

Windows PowerShell 5.1
Windows Package Manager v1.9.25200

Start by reproducing the documented winGet update command with --log and the issue's -o variant in Windows PowerShell 5.1. Trace the update command's log-option handling and compare it with the documented behavior. Done means the command creates the requested log file containing relevant update output, including the no-upgrade result.