To follow up BT,
Key switch,
Typically Three wires,
Red > +Battery
Yellow w/red stripe > Crank to slave solenoid
Purple > Ign on, goes thru large black connector and splits in harness to coil + (with a 2-3 ft long resister wire) and non resister wire to electric choke if equipped otherwise wire will be visable but not connected.
As said, issue could be at key switch, or big black connector or at split at engine harness.
Get a test light and check for voltage at key switch purple with key on.
Also egine should start when cranking but shut off when key is returned to run position based on your description. If this is not happening then look for small wire from Starter solenoid which should go to coil + which supplies voltage to coil when cranking... (two small wires at starter solenoid, yellow with red -crank for starter and I believe purple with red? Goes to coil +) What ever the color it goes to coil.