Grounding the switch wire to the frame light's up the neutral.
That's good, proves that the wiring and connections upstream from there are good.
Here's what I would do, remove the neutral switch (or use a spare), reattach the wire to it, and ground the switch body to the frame of the bike. Light should go on. Press in on the ball of the switch, light should go out.
If that works as expected, peer up into the hole for the neutral switch and make sure you can see the depression on the shift drum, and that it isn't damaged or mucked up in any way.
If that looks good, next test I would do would be to reinstall the switch but not all the way, leave off the mounting clamp and don't push the switch all the way into the hole. Now shift the bike in and out of neutral while observing the light, any activity? Experiment with various depths of the switch, at some point you should find a spot where the light behaves normally. If you do, that means that for whatever reason the switch is "too deep" when fully mounted, so the ball never springs up enough to light up the light. Why this might be, is a mystery to me.
hth
mystic_1