One potential issue is one we're very cognizant of here in Hollywood: being typecast. We considered this carefully before deciding to make our video public.
Keep in mind, we're a very early stage company. Our video reflects that, and our product video reflects the early product we have shipped (although we do hope you download and try Cuddli instead of just watching the video!). With the wide distribution and high profile that anything linked to YC is likely to attract, we may well be creating an impression with potential future partners, investors, and employees. This is likely the reason why most of the startups in this competition haven't chosen to make their video public.
What we've found is that most people really want their video to be seen and aren't super worried about getting their ideas stolen. I think YC could get mad about this, but it's a great way to meet others who made videos and to encourage them, give feedback.