Hal, what you describe can be the result of MORE THAN ONE PROBLEM, let me cover each.
1) FAULTY IGNITION BALLAST RESISTOR....In cases where the engine fires/runs ONLY SO LONG AS the starter switch is engaged BUT STOPS FIRING/RUNNING WHEN NOT......The series external BALLAST RESISTOR (between ign switch and coil) is bad or open or its wiring is bad or open, or one of its connections. Those ballast resistors do go bad and/or burn open.
2) FAULTY STARTER SWITCH.....In cases where it takes several tries before the starter motor finally cranks the engine over, the saddle mount start switch may be bad due to burned, pitted, corroded contacts, thats fairly common.....easy to replace
3) Faulty copper button/terminal contact on side of starter under saddle mount switch is burned, pitted corroded whereby even a good starter switch may not allow the engine to be cranked over by the starter motor..A burned pitted corroded copper start button on starter motor side needs to be replaced or a new start switch will also get burned.
4) The pedal and rod assembly that pushes the start switch button down may be out of adjustment whereby its not pushing the start switch button down far and hard enough.
I would start by removing and inspecting the start switch contacts and replace if they are burned or badly pitted......When its off inspect the copper button on starter to see if its badly burned or pitted or carboned over. Clean up and smooth it down nice n shiny and it may be okay if not damaged too bad...........Make sure to get a new correct start switch with the extra small side terminal and insure its wired up to the coil correctly......Inspect the Ballast Reisitor and its wiring, it has to be wired with good connectiuions and have around 1.2 to 2 or so ohms of resistacne, if its a bad/open circuit the engien will fire/run ONLY when start switch is depressed.
You likely need a new starter switch and/or the starter motor copper button (located under saddle mount start switch) needs attention,,,,,,Check starter pedal and rod linkage to be sure its depressing start switch fully,,,,,,,,,,,Check ballast resistor and its wiring for correct resistance,,,,,,,,,
NOTE I have seen a few start switches that arent mounted just right whereby the contacts arent meshing right with the copper button
INSURE GOOD BATTERY, BATTERY CABLES, CONNECTIONS AND GROUNDS ARE ALL GOOD
Hope this helps, Im sure I missed a few things if so maybe the other fine gents can add to this. Let me know how it turns out or of you have more questions
John T In the RV near Dade City Florida