I wouldn't say that the scenarios I described are "just one small area of insurance." Most things for which people buy insurance fit under that pattern -- for a small to moderate price, you buy the right to claim a large sum that saves you, or at least alleviates your position, if an improbable ruinous event occurs.

Intrinsically, the average person must pay in more than they get out. Otherwise the insurance company would go bankrupt.

Since in case of disaster the burden falls on them, they also specialize in specific forms of damage control (e.g. by aggressive lawyering, and generally by having non-trivial knowledge on how to make the best out specific bad situations).

No reason a loan style insurance company couldn't do the exact same thing.

I'd be surprised if the modern infinitely complex mazes of business regulation don't give rise to at least some such situations.

'Rent-seeking' and 'regulatory capture' are certainly good answers to the question why doesn't this exist.