How can I remove compatibility settings that are not editable from the Compatibility tab?

I found the solution here. From within the Registry Editor, navigate to:

HKEY_CURRENT_USER\Software\Microsoft\Windows NT\CurrentVersion\AppCompatFlags\Layers

From there you can find per-user compatibility settings.

In my example I found a value named C:\Program Files\Steam\Steam.exe with data ELEVATECREATEPROCESS. Removing it fixed the problem.

Alternatively, you can use the Program Compatibility Troubleshooter.

Right click Steam shortcut. Click "Troubleshoot compatibility".

1. Choose "Continue troubleshooting the problem"
2. Choose "Troubleshoot program"
3. Uncheck all boxes
   Click "Next"
4. Select "No, I am done investigating the problem - undo any changes made and clear all settings"
   Click "Next"
5. Choose "Close the troubleshooter"

For per-applications compatibility settings, navigate to HKEY_CURRENT_USER\Software\Microsoft\Windows NT\CurrentVersion\AppCompatFlags\Compatibility Assistant\Persisted. You'll have a list of all the flagged executables, and you can delete the ones you want to undo. You might also want to check HKEY_CURRENT_USER\Software\Microsoft\Windows NT\CurrentVersion\AppCompatFlags\Layers, as previously suggested.

You can also check if there are machine specific entries in HKEY_LOCAL_MACHINE\SOFTWARE\Microsoft\Windows NT\CurrentVersion\AppCompatFlags\Layers.