Assuming the two coils on the contactor are designed the same ( see if you can find the resistance spec online, i forget the nominal expected values) with power disconnected from the entire system, check for voltage at each wire with your vom/dvm, mark and pull back the wires at the contactor, and measure the resistance of each coil. They should measure approximately the same. If the coil that the auto position feeds is much higher, it may be time to replace.
Regarding the switch, contact cleaner may remove enough crud to restore it, but you have to remove it from the system first and should expose the contact surfaces.
If the chlorinator control circuit ( timer or otherwise) is providing the PUMP ON control signal through the AUTO switch position, then a defect there ( low current through) could indeed be an issue. Too low of output plus going through a dirty switch contact combined might be enough to cause the initial start failure. Rotating the switch would disturb the crud on the switch contacts until it oxidizes again.
The ON switch position should be directly providing current to the contactor coil to close the contacts on power feed to the pump motor proving no issue with the motor or that coil.
Experience is a good teacher. We do tend to jump to the most common failure in our experience for the symptoms as the likely solution/idea for an issue when we cannot directly observe the physical reality. Not knowing how these systems are typically designed, i would not have thought of the chlorinator. Hence my ask for the circuit drawing. Control timers are just another set of contacts to get dirty and a motor to fail.
peace.