The dropbox.com/s/ style URL is designed to bypass all issues with permissions. It can share any file you specify from your dropbox folder. (It doesn't have to be in the public folder. )
Using that link format I can, for example, keep all my SD debugging movies and files in a private folder, but share individual items and the only people who can see the items are those who have the links.
That style link works everywhere else in nearly every other context. (In one forum I go to they only display and play youtube videos in posts, so with the dropbox link they just show the URL. I've put in a request there to display and play videos from other sources).
It even works in discourse as a link; as opposed to a video link. Discourse attempts to load and display the link and fails. It also works in discourse to show nearly every other kind of file (the exception being AppleScript apps, which it treats as folders).
I know Dropbox doesn't have the best rep, but that doesn't mean that we can just assume that any problem with a link is a dropbox problem.
There are other ways of inserting dropbox videos into posts (none as simple to use or as good as this method), but this is still a bug that should be fixed.
Here's the message I would send to Discourse:
"There is a bug with displaying videos from Dropbox using the dropbox.com/s/ format. That is the format dropbox uses for sharing specific files, and it works in Discourse for all other kinds of files. For video files rather than display the video in the posts with the option to play within the post, Discourse only displays the video's control bar.
Is this an issue with the set-up of the forum or is it an issue with Discourse displaying videos using the dropbox.com/s/ style link?"