Games store their installation path in the Windows registry.
The Fatal: Could not open registry key message means that the path information is missing from the Windows registry.
The Fatal: Error loading plugin list: <EDirectoryNotFoundException: The specified path was not found [C:\Path\To\Old\Game\Data\]> message means the Windows Registry information is out of date.
Starting the game launcher once will create the path information in the Windows registry. The game launcher can be started directly from a file manager like Windows Explorer without Steam to avoid game updates. Exit the launcher instead of starting the game, obviously.
In case there is a still a game installed at the old path, use RegEdit.exe to delete the registry entry first.
HKEY_LOCAL_MACHINE\SOFTWARE\Wow6432Node\Bethesda Softworks\[Skyrim|Skyrim Special Edition|Skyrim VR]
If the shown game mode is wrong, start the tools in the desired game mode.
In case a game version is used that does not seem to use a registry key, or if there are multiple active installations of the same game mode in different paths and constantly updating the Windows Registry manually is not an option, use MO2 Fix Game Registry Plugin or set the path to the data folder of the game with the -d:"c:\path\to\game\data\" command line argument.