Are there any examples of a kidnapping actually being profitable?It's very easy for kidnapping to be profitable.
All you need is a state with a pathetic justice system.
Start with a police force terrible at finding the real perpetrators of crime. This is very common in the world. It is often found in poor states, which don't have much money for any public spending including police and fire. Often the states are autocratic or formerly-autocratic, autocratic states don't really care who's actually guilty. Very often a state is both (ex-)autocratic and poor.
The next step you need is a bad judiciary and a pathetic set of laws. For example, even violent kidnapping which involved mailing fingers to the victim's family, could only earn the convicted two years in jail. I am not making this up, two years at the most, by law.
Then lastly, have incredibly messed up laws which actually make no claim on the ransom money even if you were found guilty. Thus you can do two years in jail, and they enjoy the ransom you extorted. Often the victim's families are not particularly wealthy, or if wealthy not really capable of exacting violent street style vengeance. So you can relax and enjoy your cash.
This is how kidnapping can become a real sector of the economy, look into Columbia in the 80s, and lots of other places.