imho, I totally agree with being able to use video.
Worrying about the video content is the same as worrying about text replies and their probable spam/dodgy contents, SE has already several "rep" locks to avoid that kind of stuff, same logic applied to video would be just enough to avoid that kind of issues. (ie. 30rep = Kubrick badge: You can screen capture bla bla)
Also, crafting a home-brewed hosted module that could actually let us upload, embed and moderate it properly (we have enough content moderation in here to worry bout one extra bit). We are talking about GDSE, half of the content is based on visuals and "how-to's", video tbh is a must to enhance the experience.
Google Hangouts use a lovely framework that could probably be replicated to screen capture through the browser, maybe limiting the duration of the captures, and such.
Having such a big community would actually end up helping this feature, SE could launch a contest to craft the module through all the involved disciplines (communities), etc... we got the experts, whoever is in here has the time at some point, the more I think about it the more I ask... why not? we got everything :) . Maybe it is not a module to include in SEOverflow, SEBitcoin, and such, but I can see this enhancing the UX for any graphical/visual community in SE big time.
(Wish I could comment already instead of writing an "answer" but hey... few rep points still to go for that, bare with me).