I would have to say Carscw and EngineMan both have given you excellent troubleshooting tips. As far as your question can the switch turn the ignition on and off (let it run and shut it off) but still not start it the answer is "YES" the on and off is one function of the switch on the first position, powering starter is accomplished by a momentary contact as you turn the switch the furthest clockwise until it starts at which point you release the key and it returns back to the first position breaking the power contact in the switch that excites the solenoid relay that powers the windings in the starter.
This process just cleans up the engine of large wires running from the battery up to the dash board then back to the starter, back in the 60's and earlier mowers did not us a remote solenoid the actually had a manual push contact relay that had large enough wire run to the dash to handle 70 to 80 starting amps. This was simpler but the day of the push starter has come and gone. This evolution happened even earlier in cars, trucks & tractors.