Let’s get it started in here.
Well there are a few options at your disposal; the easiest of these not necessarily being the most thorough but since you say the machine shifts fine through all five forward gears (and reverse is working), we would probably just disable the neutral safety switch – which will allow you to start the machine.
To do this simply ground the wire at gear position switch located at the rear of the engine. The wire in question will be light green with red stripes. Zip tie it to machine’s frame and it should start for you just fine even without the neutral indicator.
As for why your neutral indicator isn’t working, there are quite a few options. We would start by grabbing a multimeter and checking the neutral safety switch for continuity. Set to ohms, with the wire off, and with the transmission in neutral, you should read continuity to ground if the switch is working. From there check the circuits back until you locate the faulty section.