The brighter light immediately causes a higher voltage, but not hugely higher. That's the crucial part. If you want to have an image that looks like the eye expects it to, you either need to amplify the signal (increasing the differences between high and low, both correct and incorrect due to noise) or you need to read for longer, increasing the actual sample. The latter is what the sensors used in digital cameras do.
Each photosite is not just a light-sensitive photodiode, but also contains an accumulator called a "well". As the photodiode continues to produce voltage (as it is exposed to light), the accumulator fills. If the light hitting a particular site is bright, that well fills quickly. If the light is dim, it fills slowly. When the exposure is finished, the level of the well is sampled and converted to a digital value.
Of course, in bright light, there's a lot of data, so a short exposure paints an accurate picture (if you'll pardon the turn of phrase). In low light, though, there's just not much energy to measured. If you just take a quick sampling, noise from reading the sensor and other unavoidable real-world randomness will indroduce variation as strong as the "legitimate" difference between the more full and more empty photosites, and there's no way to tell which is which.
This is what happens when you take an underexposed image and try to crank up the amplification in software: noise, noise, noise, and maybe just blackness. And any instantaneous read (without an accumulator well) would not have enough data to be useful.
Simple as that, really. Turns out that modern sensors are better at this than chemical-process film: it's why we can have seemingly insane ISO values of 25k and above. Those are able to measure finely enough that a large amount of amplification can be applied without noise becoming overwhelming. Fundamentally, though, compared to the magical instant-read device, we're still in the same ballpark.