You can use "on" in a different context, for example if you follow "on" with what he was unburdening himself about or the place where he did it.
He unburdened himself on the subject of his marriage.
He unburdened himself on the flight.
As for "on" followed by a reference to the person receiving the unburdening, I've heard it sometimes, and it's comparatively casual. I agree "to" is the best choice.