I'll tell everybody why there has been no EF Mount Video Camera so far. Lean in closely...closer...
IT'S BECAUSE IT'S A DUMB IDEA!
Really now, who thought using EF lenses for video was a good idea? Most are terrible for video work in at least three different ways, possibly more. Even the primes. I'm not going over all the reasons why, but the Lens Rental guy did a couple great articles on this, and I added a comment with my own thoughts to his latest article.
Now, could Canon make EF mount lenses appropriate for video? I suppose so, but the way many ENG / handheld style video cameras do AF is a lot different from still cameras - contast AF versus phase detection. I would not be surprised if the resulting EF mount signal line design would be somehow handicapped for video use. A video-dedicted mount should be a better solution all around as I don't think the EF mount really supports body-based zooming, i.e. the rocker switch zoom control on hand- and shoulder-held video cameras. I am also not sure if the jerky aperture changes (i.e. changing aperture from two values may momentarily open up the aperture completely, and if I remember right it may close down further, too) of a stills camera are due to the body or the lens. If it is something in the lens then this is a further problem. Same for the issue of how precise those aperture steps may be - for video (at least movie quality) it seems desirable to be able to smoothly open and close the aperture.
The one very important plus, which nobody has missed, of going with EF mount is that the image circle is much better for larger sensors. But this is just about the only plus of that system.
There is of course the question of what these cameras are to be used for. For news channels they have camcorder type cameras; Canon already has a line of zoom lenses. For movie studio type work, Canon has apparently released their own PL Mount lenses (a prediction I got completely wrong in short order).
I think that the growing popularity of DSLRs for video projects means that yes, there is a market for something better with the EF mount. But if you're going to market something as having better ergonomics and functionality for video, it makes little sense to then handicap that with the expectation that EF mount lenses will be a good solution. That said, there are some SLR lenses that would be well suited to special effects on an EF mount camera, but only a handful would really be useful for video, and only the most unusual (i.e. macros, tilt-shift lenses) would really be worthwhile for pros, and only because they have no direct video equivalents.