Why can't we apply Gauss' Law to find the electric field coming from the central axis of a uniformly charged disk of negligible thickness? If we can use a Gaussian cylinder/pillbox to find the the flux from a small piece of charge on an infinite plane of charge, why can't we do it for small pieces of charges on the disk of charge? After all, we have the symmetry of the Gaussian surface being a cylinder as a valid symmetry we can take advantage of because it has an easy surface area that well eventually cancels out and easily calculate the electric field, right?
No. The problem is that the electric field from a disk of charge looks like this:
See how the arrows point outward towards the edge of the disk? That means that there will be some non-zero flux through the wall of the cylinder, and that flux is difficult to calculate.
You might argue that the arrows along the axis of symmetry of the disk are pointing directly away from the disk, which is true; however, because your Gaussian cylinder has a non-zero radius, you will have to take the arrows away from the axis of symmetry into consideration.