Quote by alcoholicsephiroth
But what of the general definition of moment of inertia ?
I = sum of mr^2 for each particle in the solid.

That's true for particles, not for disks.
Does it not follow from this definition that the moment of inertia of any continuous solid can be found by turning the above sum into an integral, without the need to consider the rotational inertia of the infinitesimal element of choice, as I tried ?

Only if your element is a point mass. You took a shortcut by taking disks as your mass element. If you started with a particle mass element, and did the full integration, you'd get dI = 1/2 dm r^2 for each disk.