While I don't think it is impossible to produce a video camera that could produce a good still, I also have yet to see a video camera actually do it.

I think there are divers reasons as to why:

• Digital Video has, for a long time, been a low resolution format. Yes, the sensor is usually capable of much more than, say, 640x480, but what's going on is that the sum of a larger number of pixels at the sensor level is going into the output of a single pixel in the video format. This permits two things: 1) "better" light gathering capabilities of this output pixel, and 2) Increased perception of sharpness.
• This increased perception of sharpness means that the optics on the camera don't have to be all that great. The optics are going to be made to get as much light as possible in 1/30s or 1/60s (or even 1/24s) to the sensor, and then permit the downscaling to "increase" the sharpness of the image. What might look just a bit fuzzy at the full 15mp (or whatever it is), will probably look just fine at 2mp (which is HD, btw).
• Video is a continuous experience, and our vision takes the incoming video and tends to sum it over many frames. So lower-quality sensors can be quite acceptable. If the color isn't perfect in each frame, or if there's a bit of noise across the frames, etc., it really doesn't matter to a large degree primarily because our brains will average it out, and so as long as it isn't overtly horrible (or long in duration), our brains will take care of it. But look at a still from the recording, and it's going to be horrible. (To some degree, I think this is why we can accept video compression; as long as it isn't so obvious as to cause obvious blocking and other visual artifacting, our brains are happy to accept it as "close enough" and go with it.)
• While the zoom range is quite nice (30x, 60x, 80x, etc.), you're dealing with an incredibly small sensor, and the fact that to get a stable image you need to have a shutter speed of at least 1/, which means high ISOs. Now look at a compact digital camera using a small sensor at ISO1600 and 300mm-equivalent. That image will look nasty.
  • Side note: whilst you could tell that digital camera to go back to ISO100 and hold the shutter open for a lot longer, there are many video cameras that aren't going to permit that option. Some do, but don't count on it. Which means even if you get that tripod, it's going to shoot at high ISO regardless of what you think it should do.

Now, professional level digital video cameras (like those used by major studios), are probably pretty good at giving the control necessary to get good stills. But you're also looking at good glass in front of the sensor, and the level of control is much higher as well. I'm sure the RED Scarlet does fine, but it also costs a very pretty penny, and may as well be considered a hybrid system that can do both photography and video.