I get the “not authorised” message with the nsfw link at the bottom of the Flicking Channels video. Which I presume is just a link to a folder and the default file within that folder is either a video or a page with a video embedded into it.
In the forums, most of the videos are hosted on vimeo, so they’re not being played from the age restricted folder. They’re being played from a folder on vimeo.
So far, the only video I’ve found which is actually played from a folder on NE is the Horizon video in What’s the Problem with Nudity but with that, only the forum page is hosted within the age-restricted area. The video itself is not stored in the same age-restricted folder.
My guess is that whatever controls the age restriction is doing it by controlling access to the forum pages. But it seems not to be giving its protected folder permission at the operating system level, which is what would be needed to allow direct access to the video from a correctly logged-in browser.