Everything in the electric start system focuses on one thing - the "neutral switch". It isn't really a "switch". It's actually a plastic and metal plug under the countershaft sprocket cover with a metal circuit running through it from a wire on the outside to a contact on the inside end.
When the shift drum is in the neutral position there is a tab on the end that makes physical contact inside the engine with the neutral switch end which provides a ground to the wiring harness (LIGHT GREEN wire with RED STRIPE) through the engine to the frame.
The neutral switch is the ground at the end of the circuit that triggers the starter relay. This creates a safety interlock to prevent runaways when starting in gear. Engine will only start in neutral.
Starter circuit Description.
Battery must be fully charged and functioning properly.
Ignition switch on sends battery power to the low tension side of the relay. Pushing starter button with the bike in neutral completes the circuit from the relay to ground and the relay should close.
The closed relay connects the battery positive high tension cable across the relay posts to power the starter motor, which is grounded to the engine/frame - no negative wire. If the neutral switch ground isn't effective (worn contact, broken internal circuit), the wiring is frayed, shorted or disconnected, the starter button is corroded, the relay is sticky or the ignition switch flaky - the relay usually won't close reliably and you'll have the starter problems that are typical of this model bike. Neutral switches fail. They are made from plastic that can become brittle over time, the metal parts inside will break or otherwise short out, or someone may break the switch while working on the bikeand not care if the starter doesn't work, leaving it for later. Now it's your problem. Lucky you.
If the neutral switch is dead - replacements are available domestically and abroad. If your battery is charged and otherwise fully functional and your neutral switch tests OK and the electric starter still doesn't work, the most common faults that remain are 'sticky' starter relays, and inadequate wiring quality between the switch, starter button on the handlebars and the relay.
Simple answers before you get too involved with the search. If the battery isn't charged - you're wasting your time with any searching.
Put a charger on the battery. If it won't hold a full charge - replace it.
If the battery is good and you see a bright neutral light in the speedo - you more likely than not have a good neutral switch.
If you push the starter button and the relay just 'clicks' and nothing spins - either the wiring isn't carrying the ground to the handlebar switch, or the switch isn't transmitting the ground to the relay, or the relay isn't loose enough to close with the current and ground available to it.
The handlebar switch contacts might be corroded, or the relay sticky from lack of use. Bench firing the relay a bunch of times on the battery without the starter as a load should free it up enough for the relay to be eliminated as a fault.
If the relay is working and the starter motor still doesn't spin, make sure the transmission is in neutral and carefully short the two big posts on the relay together (insulate yourself from the big electrical arc it will create - and don't use anything like a favorite screwdriver, as it will be eaten like using a welder on it). Shorting the two connects the positive connection to the motor (ground is to the engine and frame - no wire). The motor should spin.
Starter motors, their drive chains and clutch are virtually indestructible. They are the least likely places to find a fault. If they truly are damaged or faulty, it will be a little work to fix or replace them, but new motors are available and the drive can be fixed with the removal of the generator.