If you're looking ONLY at video, you don't need to go with L glass. There are plenty of other f2.8 tele lenses in this length - namely something like the tokina 80-200 f2.8 atx-pro being a solid option for 1/5 of the price on the used market.. (slightly more chromatic aberration wide open than the L lenses, but you can't really tell that shit in video anyway)
The thing I dislike about L lenses (and really any USM lens with video) is that there are no 'hard stops' for infinity and closest focus distances.. meaning I can accidentally focus beyond infinity on a subject and fail to get them in the right focus spot. If you're only shooting video, look in the direction of vintage 80-200 f2.8 or f4 zooms - they'll be lighter, and have better ergonomics for focusing with.
Also, while f2.8 might be sexy, I'd really only recommended if you're going to throw a 2x teleconverter on it at any point - which would make it a 140-400 f5.6 lens - an f4 lens would turn it into a 140-400 f8 lens which isn't the best for situations when the light is optimal, yet dimmer.