Is that also an external boot drive? I see the System Folder and other items on the root. If so, that may be part of the problem since that would break the ability to boot the external drive since the root of a macOS 10.15+ boot volume has been read-only (and signed & sealed since 11.x+).
An external data drive should have "Ignore Ownership on this volume" checked, but you do NOT want to do that if it is also an external boot drive.
The greyed out/ghosted folder means you don't have permissions which is confirmed by the screenshot from Get Info. It should show you as owner, but you should be able to read the contents unless some other ACL has been set on that folder.
You need to create a new APFS volume for storing data if you want it outside of the "/Users" folder of any boot drive. Then maybe you can select "Ignore Ownership on this volume" without it affecting the ability of the OS to boot on that external drive.
Your current method is going to lead you into a lot of trouble & a huge mess someday if it hasn't done so already.
I hope you have a good backup of your main boot drive and the external drive including the items outside of the home user folder(s).