I'm in the same situation as the original poster - ie. the "Open Link in New Private Window" operates as "Open New Private Window" (without opening the link)
I can confirm that when creating a new user account, everything works fine, so it must be something user related. However, it doesn't appear that com.apple.safari.plist is used anymore - it has next to nothing in it and it doesn't appear to be accessed when Safari starts up. Deleting it has no noticeable effect on anything (including preferences).
On a lark, I made a local snapshot then deleted every Safari file & directory under ~/Library then logged out and back in. This wiped out a lot of Safari preferences (except for iCloud Safari shared settings/history etc.) However, the "Open Link in New Private Window" still failed to work. After experimenting for a while and not having any success, I restored from the local snapshot to get things back to normal.
I'm stumped as to what file or setting is causing this bug.