Could a camera record sensor data continuously during an exposure for more flexible post-processing?

A digital sensor isn't really best described as "reading data". A much better way to describe it is "collecting photons" that are then converted into data by measuring the microscopic electrical charges they produce once the collection period is over. They do not have the capability to continuously record the changing state of each pixel well as they collect light. And depending on how little or how much light is falling on the sensor it might take a long time for enough photons to strike the sensor before anything more than random data is generated. On the other hand, in very bright light sometimes all of the pixel wells can fill up so fast that any additional photons falling on the sensor are lost.

In the first scenario not enough photons are collected to create a discernable pattern through the "noise" generated by the energy flowing through the sensor that is used to collect the voltages created by the photons falling into the pixel wells. Thus no usable information is collected. Your entire photo is dark with random spots of color and light.

In the second scenario so many photons are collected that every pixel is read out at the same maximum value, called full saturation, and since every pixel in the image has the same value no usable information has been preserved. Your entire photo is solid bright white.

It is only when enough photons strike a sensor that the areas with more photons per unit of time have a higher readout value than areas with fewer photons striking them per unit of time. Only then has the sensor collected meaningful information that can differentiate between areas of varying brightness.

Imagine setting out a number of water buckets in your yard to collect raindrops. Imagine that they all have some water in them but you dump it out before you place them. Some are placed under the eaves of your house's roof. Some are placed under large trees in your yard. Some are placed out the open. Some are placed under the spout that dumps the water from your gutters into the yard. Then it starts to rain.

Let's say that it only rains for a very short time: 15 seconds. There are a few drops of water in each bucket. But there's not enough water in each bucket to be able to tell if each bucket may have had more rainwater fall in it or if it may have just had a few more drops left in the bucket when you dumped the water out before you put the buckets in the yard. Since you don't have enough data to be able to determine how much rain fell on which parts of the yard, you dump all of the buckets out and wait for it to rain again.

This time it rains for several days. By the time it stops raining every bucket in the yard is overflowing. Even though you are fairly certain some buckets filled up faster than other buckets, you have no way of knowing which buckets filled up quickest and which buckets filled up last. So you need to dump the buckets out again and wait for more rain.

On your third attempt it rains for three hours and then stops raining. You go out to the yard and inspect your buckets. Some are almost full! Some have barely any water in them at all! Most have varying amounts of water in between the two extremes. Now you can use the location of each bucket to determine how much rain fell on each area of your yard.

The reason we alter exposure in digital cameras is to attempt to collect enough light that the brightest areas are almost, but not quite, saturated. Ideally this occurs with the camera at base ISO sensitivity. Sometimes, though, there isn't enough light to do this. Even at the largest aperture available we can't collect enough light in the longest amount of time we dare leave the shutter open (due to motion of our subjects). What we do in this case is adjust the ISO setting in our camera so that all of the values coming off the sensor are multiplied at a factor that brings the highest values to a point where they are almost, but not quite saturated. Unfortunately, when we amplify the signal (the voltages created by photons landing in pixels wells) we also amplify the noise (the random uneven voltages produced by the current used to collect the voltages from each pixel well). This results in a lower signal-to-noise ratio which decreases the amount of detail we can create from the data we have collected from the sensor.

There are other technical limitations that prevent cameras from keeping a "running total" of the number of photons collected at various intervals while the shutter is open. Throw enough money at the problem and some of those limitations can be overcome, at least partially. But either the laws of physics would need to change or we need to completely change the way sensors count photons before others of those limitations could be overcome. Eventually the technology in some or all of these devices might replace the way we currently capture very high quality images, but we're nowhere near there yet.