First of all, if the motor starter assembly is UL-listed, its internal wiring is not in violation of NEC by definition. The scope of the NEC is limited to field installation, and does not include internal wiring of listed equipment. The NEC would apply to remote devices from the starter. See also NEC 430-73.
The location of motor overload contacts is covered in the NEMA Industrial Control System ICS-2 Standards. The reason the contacts are typically on the neutral side of the contactor is to protect them from high fault current that could occur if a remote device in the control circuit became grounded. The high current can cause the normally-closed contacts to weld shut. This would not be detected until it was too late.
I do not think addition of a light is good idea. Overloads can be obtained with auxiliary contacts as well as positive mechanical indication of operation. I'm not quite sure what the push-to-test light would really tell you. If the motor is running, the OL contacts must be closed. If the motor won't start, the OL relay is an obvious place to check.