But on my Admin account (and on my Admin account in Safe Mode) there is no C:\Users\All Users\Desktop - when I try to go there it says it's not accessible and access is denied. When I type in just C:\Users\All Users I do not see Desktop - I even tried viewing hidden folders. Nothing.
Edit: Despite all that, when I try and create a "Desktop" folder in C:\Users\All Users it tells me it already exists. It asks me if I want to merge the folders.
Edit 2: SOLVED! Thank you for your help, TorrentG, without it I wouldn't have been sure the type of things I should be trying to do to gain access.
What worked was first, making it so I could view hidden folders (Control Panel > Folder Options > View tab > Show hidden files and folders) then I went into C:\Users\Public\Desktop (which is a hidden folder) and cut and pasted the Games folder - not copied - if I copied it then I would end up with the folder twice on the Admin Desktop, with no way to delete the first one, as it would have deleted the copied one as well.
I still don't understand the whole thing with the Desktop folder in the All Users section, but in any case, I've found a way to fix the problem I had!