In Music.StackExchange, and StackOverflow, there is a feature of embedding videos from YouTube and other popular sites by just placing URL in a new line. There are plenty of Islamic videos on YouTube that can be used to enhance an answer or a question. Should this feature not be available at this site too?
The primary advantage to embedding video is that it allows you to both see and hear information, rather than merely read it. While I admit they can be full of valuable knowledge, I suspect the majority of Islamic videos which would end up being embedded would be speeches; an eminent scholar or a respected imam explaining a point in detail.
The thing about speeches is, they are not significantly improved (from a Q&A perspective) by being on video. Words can easily be captured in text form, and (much like essays) the entirety of the video may not even be directly relevant to the actual question, necessitating a summary rather than a verbatim transcript or a complete viewing.
The only situations I can think of that would actually benefit from a video embed would be anything that would require a visual demonstration (e.g., "How to perform salat?"), or anything that would require an audio demonstration (e.g., anything involving tajwid). However, given both the relative rarity of these questions and the potential for abuse, I feel that leaving video as link-only is the better option.
My biggest concern about videos is that they are not searchable so cannot be easily found. When a video becomes a substitute for a well-written answer, that answer no longer provides a point of entry for people searching for that information.
@goldPseudo mentioned videos of speeches. Not only would those videos not enhance the answer, but they would make the content virtually unfindable. Remember that search is the lifeblood of this site, and videos are essentially a black box to the Internet — they break many of the major mechanisms that make the Internet work.
Like it or not, text is the primary means of communicating what we have here to the outside world. For every answer that is done "in video," that information becomes obscured behind a black curtain. The nuances of what should be video and what should be typed out will be lost on most users. Every time a video is posted, there will like by one less opportunity for people to find the information contained inside that video.