Exactly this happened to me with a BMW I once owned.
Don't be too dismissive of the kill switch as being the culprit.
That was the case with my Beemer.
Yes I know the engine won't turn over with the switch in the off position, but a dodgy connection in mine was causing the engine to cut but still allowing it to turn over on the starter.
The switch was sort of stuck between the on/off position.
By the way, what bike is it.
I don't think modern Italian bikes use Italian electrics.
I think they use Bosche, which incidentally, is what BMW use.