They do that based on a misapplication of the divergence theorem. They assume that all the mass inside an orbital radius can be treated as a point-mass which is true for uniform spherical distributions. They also assume the mass outside the radius has no net infuence - which is true for uniform spherical shells. Neither of these is valid for non-uniform distributions - a flat disk in not symmetric in 3-space. If they'd just do some electrostatics problems they'd know the different formulas for disks and rings. I was going to google a sample page but there's too many to sift through and I gotta run an errand.
Welcome to the club - it's incredibly small...