Why don’t image sensors record exposure over time for built-in HDR in a single shot?

It's been done in X-rays.

The TimePix is a 256x256 detector. It has three operating modes:

• the usual "total energy in this pixel since we started integrating";
• Time-over-Threshold (TOT): the detected pulse height is recorded in the pixel counter in the TOT mode; and
• Time-of-Arrival (TOA): the TOA mode measures time between trigger and arrival of the radiation into each pixel.

This technology has been adapted to optical imaging. The TOT mode is best described as acting like a Wilkinson ADC -- the readout corresponds to total time where accumulated charge is at or above threshold. Subtracting this from the shutter time tells you how long it took for this pixel to saturate. Thus, for each pixel, you can draw the line from 0 to saturation over time since the shutter opened. So you can pick whatever virtual shutter time you want (as long as all the pixels saturated) and use each pixel's line to estimate its accumulated light up to that virtual shutter time.

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A more direct implementation of your idea has been done in CMOS. Each pixel records and reports its time to reach a threshold charge. (Rather than ADC the pixels that do not saturate in time, the threshold is swept, so every pixel eventually exceeds a sufficiently lowered threshold.)

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I recall that the Pixim Digital Pixel System (example) also did this using an ADC per pixel and nondestructively reading out the accumulated charge repeatedly (to get the accumulation slope). But I can't find current corroborating evidence.