You added private planes to try to make a profit? There's your problem right there. They're not supposed to.
Mooney Bravo is available as cargo type. And in 96comic they are very useful for mail and not-so-profitable cargo. They are also very cheap and have a very low cost/km.
I think that the oil rig is the only place to use helicopters. Even if they are not (very) profitable, the passenger boost should be worth it.
I do think too. Even on real life they are hard to find. I always lived in cities with <500,000 inhabitants and hardly saw one. Sometimes when I visited some cities around here with 3,000,000+ I hardly saw one. Most of the times it's the police or the hospital helicopter.
I find interesting add helicopters in Simutrans, but I can't see a good, profitable and somewhat realistic way to use of them.