Others have already explained why this won't work, technically. I want to touch on why it wouldn't work practically.
If data storage were not an issue, is there any reason this couldn't be the norm, at least for professional and art photography?
Consider the magnitude of different lighting conditions that we may want to take photographs of. Even ignoring extremes such as astrophotography (where you are often photographing small speckles of light surrounded by nearly total black), you still have evening or night terrestrial photography, and brightly lit snow-covered winter landscapes. I'm going to use the latter two as examples.
Also, I'm going to assume that in order to accurately recreate any desired exposure, we have to expose the sensor to the point of full saturation.
Also, I'm going to assume that we can read the sensor values in a non-destructive fashion. (This is probably one of those problems that fall into the category of "throw enough money at the problem and it might be solvable".)
In the night photography case, we would need to expose the sensor for a very long time to saturate all pixels, which means that any photo, no matter what we actually want a picture of, is going to take absurdly long to take. The classic tourist picture of dancers at an outdoors bar becomes nearly impossible because, well, you might be able to snap a few of those during an entire evening. Not good. So we can't expose to saturation, at least not indiscriminately. (Exposing to some percentage of pixels being saturated is equally useless, but for different reasons; try getting the exposure exactly right when taking a photograph of a fireplace with a fire burning in it. That's almost impossible; no matter how hard you try, some pixels will be overblown or huge swaths of the image will be horribly underexposed.)
When photographing a brightly lit snow-covered landscape, such as a winter vista during daytime when the sun is out, the exposure that the camera's automatic exposure system aims for ("18% gray") is woefully inadequate. This is why you often see photos of snow that are dark, and where the snow appears more of a light gray than white. Because of this, we often use a positive exposure compensation setting that results in the snow being exposed as a nearly saturated white. However, this means that we can't rely on the camera's AE system to determine when to end the exposure: if we do, such pictures will invariably be underexposed.
In other words, exposure to full saturation is impractical in many cases, and exposure to make the AE system happy is inadequate in many cases. This means that the photographer will still have to make some sort of choice, and at that point, we are at least just as well off staying with what we have and photographers are used to, making the AE systems better and giving the photographer easy (easier?) access to exposure compensation settings. By increasing the sensor's practically usable dynamic range, we can allow (even) greater latitude in exposure changes in post-processing; the original digital SLRs were horrendeously expensive, yet truly horrible in this regard compared to today's even entry-level models.
All of which can be done fully within the framework of what we already have. This is not to say that dramatically improving the usable dynamic range of the sensor is easy, but it's probably a lot easier than what you are proposing, and it's a problem vendors have experience working on.
Professionals, almost by definition, know how to use the equipment of their trade. It isn't really any different if they are photographers or space shuttle pilots. Especially when it can be done without causing information overload, it's usually better to give the user full control of professional equipment. In my opinion, current high-end DSLRs are pretty good about hitting the sweet spot on this.