The problems I see with hosting videos, is that they stop being available all of a sudden if they come from YouTube. So, you just have to keep updating the page and either hosting the link if the enbeded video is not available anymore.

Exactly. That's one of the reasons that I listed before. There really is nothing worse as well than having a dead link for a video. Even if you're attentive to your page a dead video link embedded into the page will make it look like the admin doesn't care about keeping up on the page's functionality.

Unless of course you are posting your own videos. One of my friends told me that he made copies and downloaded the videos into his computer, renamed them slightly, gave the credit to the original poster and would have them on YouTube but him as the author, so he could control it to be available.

So far I have been fortunate. Videos that I have linked out to have remained available. Of course, these are on my Squidoo pages so there may be some special code that keeps hunting for redirects periodically.