Can break System Restore. Why not fix that first? It could be why this happened.
As to other fixes I know of, it required a registry edit but your security software borked System Restore (well discussed area so I'll leave it at that.)
If you use Glary or similar registry cleaners along with a long list of other things you can see the right click new fail. At least the old command line MD works and most of the time the File, New menu (not the right click but the menu) works.
Given the handcuffs you put on fixing I don't think I can help, only point out what may be what broke it. And you don't want to remove the cause or use the usual fixes.
I can’t create a new folder or new file in Windows 10 file explorer. When I right click the mouse and click “new”, it just hangs. And in the file explorer, when I click on “New item”, it doesn’t do anything at all – doesn’t hang and just doesn’t do anything.
I searched the internet for some advice and tried several things:
1.sfc /scannow
2.stop the Windows Explorer task in the task manager
3.run “dism.exe/online/cleanup-image/restorehealth
None of these things worked.
Another said to either delete or rename this registry key: \HKEY_CURRENT_USER\Software\Microsoft\Windows\CurrentVersion\Explorer\Discardable\PostSetup\ShellNew
One said to do a System Restore, but it aborted saying my security software – BitDefender needed to be removed.
I don’t want to mess around with the registry or my security software unless I know it will work.
Does anyone know how to fix this problem?

Chowhound
Comic Vine
GameFAQs
GameSpot
Giant Bomb
TechRepublic