I have fiddled around with GPO and some registry, here is what I have found so far:
GPO proved to be useless. There are settings regarding hiding the system icons, however they only work if the icons are enabled. If the clock is already hidden due to Win11, setting the "Hide Clock" to Disabled or unconfigured does not enable them, same with the other icons (but it does hide them if they are enabled.)
By deleting IconStreams and PastIconStreams, I had hoped that Windows would re-create them with default values - which show the clock - but with no luck.
In HKCU\SOFTWARE\Microsoft\Windows\CurrentVersion\Explorer, I tried to create d-words "HideClock" and setting it to 0, but it had the same issue as editing it in the GPO.
Out of desperation, I have enabled the icons, then exported the entire Explorer registry tree (HKCU\SOFTWARE\Microsoft\Windows\CurrentVersion\Explorer), disabled the icons, loaded the exported registry and restarted Explorer. To my surprise, the clock was back! I incrementally deleted entries and loaded the .reg until the clock did not show. I do not recommend this method, as it messes with icons pinned on the taskbar, and I'm sure there are some side effects I cannot see immediately. But if you still wish to try this out, the registry keys that remained and worked for me were:
HKCU\SOFTWARE\Microsoft\Windows\CurrentVersion\Explorer\Streams - Settings
HKCU\SOFTWARE\Microsoft\Windows\CurrentVersion\Explorer\Streams\Defaults - {24CCB8A6-C45A-477D-B940-3382B9225668}
HKCU\SOFTWARE\Microsoft\Windows\CurrentVersion\Explorer\StuckRects3 - Settings
I am importing the settings from a .reg file through a .bat file that I placed in C:\ProgramData\Microsoft\Windows\Start\Programs\StartUp, so that it launches on every user login (admin privileges are not required, runs fine under any user account).
In the batch file, I provide the path to import the .reg file, kill and then start Explorer.exe for the changes to take effect.
So far, it looks like we will be discontinuing roaming profiles in favor of local profiles with folder redirection.
It is an absolutely terrible "solution" until every PC in the company runs on Win11, but I have not gotten anything else to work unfortunately...