First, you know that the battery is not part of the ignition system? The engine does not need a battery to run. The battery is just to power the lights and electric starter motor.
Power goes to the key switch via the red wire - when the key is turned to the on position, the black wire becomes 12v+ - the black wire goes to the neutral and reverse lights - a green and a light green wire with a red stripe goes to the neutral/reverse switch - when the lg/r wire is grounded the neutral light illuminates - when the green wire is grounded the reverse light illuminiates.
Now the lg/r wire goes to a diode - on the other side of the diode is a light green wire - the lg wire goes to the CDI unit and the starter solenoid - the starter solenoid has no ground until the neutral light is grounded - fix the neutral switch or the wiring to it and it will function as designed again.
Power goes to the key switch via the red wire - when the key is turned to the on position, the black wire becomes 12v+ - the black wire goes to the neutral and reverse lights - a green and a light green wire with a red stripe goes to the neutral/reverse switch - when the lg/r wire is grounded the neutral light illuminates - when the green wire is grounded the reverse light illuminiates.
Now the lg/r wire goes to a diode - on the other side of the diode is a light green wire - the lg wire goes to the CDI unit and the starter solenoid - the starter solenoid has no ground until the neutral light is grounded - fix the neutral switch or the wiring to it and it will function as designed again.