Ah , there's a hint. So when the starter is working and you hear the engine trying to start, that's consider "turning over"
Not too many cases of these engines running rich and I would put that lower on the list for now. The rolling back and forth and then it works is kind of strange and makes me wonder if it's a safety switch or key thing but it may just be coincidence
I would call your issue a "Crank no start " which does appear to happen quite often . Some people mess with the EFI relay a bit to help diagnose. I could be way off but when I hear " the engines turning over but not starting " this is my usual advice.
Some light reading material
EFI Relay Bypass Kit (Immobilizer Fix) for UZJ100/LX470 - https://forum.ih8mud.com/threads/efi-relay-bypass-kit-immobilizer-fix-for-uzj100-lx470.952811/
If you're engine is turning over but not starting keep the 4 basics in mind. You need fuel, air, spark, and compression. Chances are air and compression are fine unless you recently overheated or have a rat living in your airbox..... Focus on fuel and spark.....
Fuel delivery could be an EFI relay or pump issue. Spark issues could be from a bad coil but you should be able to get the motor to fire up even with a dead coil. I'm not sure what the neutral safety switch will disable.
This thread talks about nuetral safety switch symptoms.
Could also consider basic maintenance stuff like cleaning the MAF and resetting the battery for a fresh start.
Just my guess