Let's be clear, YouTube's TOS does not grant you the right to modify, download, or use third party tools to increase the functionality of the service. The use of tools such as KeepVid certainly fall outside YouTube's TOS. In short, Google is covering their legal butts, any legal issues (Warner Brothers) will fall flatly upon you and the schools you represent.
That being said, there are a host of good reasons to download YouTube videos you do not own. You may be giving a presentation where wifi access is sketchy or work for a school district whose broadband is maxed out. There is always that just in case scenario of if something can go wrong with wifi, it will.
KeepVid seemed the most straight forward of the download services I tried. Many were reliant on Java or just posed too many hurdles to jump through. KeepVid used to require Java to work, now it doesn't which means it works with Chromebooks. All you do is put in the YouTube url, and click download. You will then have different video / audio download options.