In short, I'd ask you this: what, precisely, of the following chain of events is substantially different between still and video? And which of that is not already a requirement of consumer cameras?
1. Expose sensor
2. Read data from sensor
4. Encode a frame
5. Save to media
Really, I don't get it: why is video not just a (nearly) free bonus?
I am no engineer.
But as I see it:
For video, a sensor with 2 megapixels (FullHD - 1920x1080) is all that is required. Videos are viewed on monitors or on screens/beamers - all of which offer - at best! - Full HD.
The sensor+readout+processor needs to be able to handle an ongoing massive stream of data without any interruption. Image quality of single frames however is a secondary requirement at best.
The sensor needs to be of a type that can handle capture incoming light for "indefinite periods" of time. Cooling that thing is a major hardware issue.
These requirements preclude certain sensor types (e.g. CCD-FT) from being used in regular video-enabled cameras altogether. This narrows the choice of image capturing device and layout friom the start to a much narrower field than for "stills capture only".
For photography (stills) hat is needed/wanted?
Sensors with the highest possible resolution [currently 36 MP+ on 36x24mm "FF"sensor size], with the best possible S/R [translating into good DR, low hi-ISO noise, and all other goodies us photographers want). Image quality of each single frame/capture is paramount, speed is a secondary concern - 10 fps more than good enough for virtually anything. Exposure times are typically fractions of a second, typically max. 30 seconds, and only in very rae instances minutes. Cooling is much less of a concern than in video use ... and yes, live view has is blurring that requirement a bit. But bottom line: the whole data readout/processing pipeline needs to be geared to highest "single capture performance" vs. "streaming performance".
These two sets of requirements are not 180 degrees opposite of each other, but there is a significant rift between them. Fulfilling both requirements necessitates an enormous amount of compromise on both ends .. for video and for stills capture, making the final product significantly more difficult to design, test, manufacture = more expensive, more prone to defects, less good in each of the 2 disciplines.
"Video" in DSLRs of all things with all the mechanical stuff (anything inside the mirror box) and a lot of the optical stuff (prism, viewfinder etc.) really *in the way of video* rather than complementary is an aberration in camera design. Looking at mirrorless cameras changes the pictures a lot, but for DSLRs its madness.
The sole reason HDSLRs got popular with videographers is their relative pricing compared to the "traditionally outlandish prices" for (relatively) large-sensored videocameras. All of a sudden, a $ 2,000 body comes with a sensor that rivals old-school videocams at $ 100.000+ ... that is the appeal in HDSLRs. It is a boon for videographers for sure, but not for photography/stills capture! HDSRLs are highly affordable to a large group of aspiring moving images people, and given the price they are more than illing to put up with all of the disadvantages HDSLRs bring to their work. Basically, all they use in a HDSLR is the sensor and the data processing pipeline. They (generally) don't want AF, but will rather add "follow-focus" contraptions and all sorts of bulky rigs around the poor little HDSLRs. They don't need an optical viewfinder (with heavy/bulky, costly glass prism). Basically, 50% of the cost of an HDSLR is "wasted" on them. Or, put another way, the money put into AF, mirrorbox, prism, optical viewfinder should go into even more video-suitable sensors+data processing capabilities. No reason, why there should not be 2k video cams with an FF sensor and an EF/PL mount in front. Without AF system, no OVF, but top notch EVF. In one word: a videocam! Something like a Canon C300, but at $ 2,000 not at an insane cost of $ 16,000 $
At the same time stills shooters should get a 5D III with e.g. a 24 MP FF "stills-optimized" sensor [whatever type/design may be best for that], top-notch AF, top-notch optical viewfinder minus all video-related features for $ 2,000 or less.
I guess that should answer the question, whether video-capability is just "a little freebie" on a HDSLR. It is not. It is a product design aberration from the very start!