Couldn't we just blur the result of the shadowing?

I think blurring the result is a way to "denoise" the image, but here we are just seeing the default result from the renderer.

Is it possible that in low light situations, we might desire this sort of behavior? From my understanding, photons hit in a "shot noise" pattern because of their Poisson distribution, resulting in a very salt and pepper image in low signal-to-noise ratio situations. Maybe this salt-n-pepper image could perhaps be desirable?

@jchh your right that shot noise is the predominant noise of photons, but that is the same as some types of current through diodes as well. This shot noise exists on the level of the individual photon or electron, and typically the "salt and pepper" we observe in high ISO is the error introduced when we gain up the image in low light conditions; the shot noise is all but averaged out since there's so much of it.