I think , if the "click" you are hearing is the solenoid , then the switch is working ....right ?
If the solenoid is closing , it is sending power to the starter , which just isn't working.
I've seen starters act that way before. If the brushes are shot , they just don't make contact sometimes , which is why tapping starter with a hammer helps .... sometimes , the vibrations probably move the rotor just a touch and you get contact.
Now , since you have cleaned the cables , are you still getting smoky wires ?