I'm using Microsoft powertoys keyboard manager to rebind some keys, and they don't work in the game since it is being run as administrator. If I do the suggestions in this forum to go to the shortcuts compatibility mode and uncheck run as Admin, then I small laucher window pops up, but once you click "Launch game" nothing happens.

Does the game strictly need windows 7 compatibility mode and admin rights?

After first laucnh it is not need adminrights strickly. And no need to any compatiblity modes (and probably even not needed fullscreen optimisation disabled pre-set by GOG).
But its unclear what the issue - with game or with "manager"...?

Also, did you install GOG Galaxy (start it as admin instaed of game exe adminrights) client game build or standard offline one (and did you start it as offline exe or still by using Galaxy?)

Witcher.exe ALWAYS requests admin mode so if it is not needed how to stop it asking for it?

One can disable it through compatibility settings not sure about negative effects though (activating compatibility mode only for local user will keep asking for elevated rights through UAC)....

No idea why they still insist on Admin rights, not even when the game installed updates into the user profile this should have been necessary.

Using Windows 11, I run the game with the satandard settings, both Win7 compatibility and admin rights can not be deactivated.

I ran into the same issue recently. Joy2Key did not work in this game, because it was running as admin & Win7 compatibility mode. Changing the game to not run in compatibility mode fixed the issue so I can play the game with a gamepad now.
BTW: I think I had enabled those settings myself, but forgot about them until now.

Here is how to disable admin and compatibility mode for applications (and The Witcher 1):
Open the properties and open the tab "compatibility", as shown in neumi5694's last picture. Click on "change settings for all users". A new window opens (see attached picture below). Here you can change the dropdown for compatibility settings and disable (uncheck) admin mode.
I launch the witcher.exe inside the "System" folder in the game installation directory. No launcher.exe needed to play the game.

The game starts and I am able to save, without running the game in admin mode nor compatibility mode on Winodws 11.