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I'm not sure I'd be able to figure it out easily if I didn't already know how a CAT scan works, but I agree it's a excellent demonstration of the principle

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Not knowing how a CAT scan works, beyond a vague intuition from already knowing what the acronym stands for, and not wanting to look it up before making a guess: can I assume it’s akin to this, in that radiation is shot from an emission source, through the target, to a detector on the other side, with the density of the material attenuating the signal in predictable ways, from which we can apply a transformation to some large number of such measurements to achieve a computed 3D result?

indeed.

In a closer analogy to a CAT scan, each nail would have a string connecting it to a ~10 degree spread of nails on the opposite side, and each string would be grayscale.

The real physics is quite tricky, because you actually emit a variety of x-ray energies at once, each with different penetrating power, and xray flux varies spatially as well. this means each of your angular strings has its own grayscale response curve.

heres a simple rundown: https://radiologykey.com/computed-tomography-16/

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