I would LOVE it if DB became a YouTube competitor. I don't share many videos on YT because it is so annoying to upload them. If I could just fling them in a DB folder and know my online video library was growing, my family and I would be posting many, many more.
However, I'd understand why DB would stay far away from that (too distracting from the core mission). They have to find somewhere new to grow into, though.
I'd look into building an app to do it, but they limit apps to 150MB files.
Even though there's an upload limit, it might still be possible to create an app that plays back uploaded files by getting a /media link and embedding the URL in a flash/HTML5 player.
A browser extension/desktop program could possibly add a "get video link" option to the Dropbox interface.
Is it really a good idea to open multiple fronts against services which are 'pro' at what they do?