Normally the clutch lever actuates a small switch when pulled. So the engine can be started if either the gearbox is in neutral _or_ the clutch lever is pulled. So i think you want to check that small switch at the clutch lever at the left side handlebar.
Bye,
Georg

1/ lots of japanese bikes have the "starter will only work in neutral with the side stand up" thing going on, it is supposed to be a safety feature, but anyone that dumb doesn't deserve to be riding a bike in my opinion, anyway you can bypass these switches, either temporarily to test or permanently.
2/ as for the finding neutral thing, assuming the clutch is ok, yes these boxes do have a tendency to go straight from 1st to 2nd or vice versa rather than having a "positive" neutral in between, _but_ if you are having peoblems then as always with any chain bike you need to closely inspect your chain (which means removing it) and check the chain tension too.
A worn or dirty chain or incorrect chain tension will always show up as worse gear changes and false neutrals and hard to select neutral etc, quite apart from making the ride generally poorer and making the engine feel rougher, it will also wreck your miles per gallon of petrol.
I've been riding nearly 30 years now and chain and tyres and two things that get DAILY checks and inspections, and by nature I'm pretty lazy leave it alone if it works kind of guy, but chains really are way up there with tyres when it comes to importance and effects on overall bike handling, performance and safety.
top tip, to check a worn chain off the bike (because only when it is off the bike can you tell things like gunk and shit making the links sticky etc) it to hold the chain out and check for bending sideways, eg at 90 degrees to how the chain bends when it is going around the sprockets.
cheers

I think you want to look at:
https://www.tr1.de/cgi-bin/service.cgi?key=xv1000&lng=eng&flg=0&pge=0042
If the starter works with the gearbox in neutral, i think that the relay nr 37 works. So it's possibly only a broken wire or disconnected connector?
Bye,
Georg