> plus a small processing fee

Okay, but small is relative.

For example, an insurance company has to pay for investigators, the number of which would likely scale linearly with customers.

Nice offices and an advertising presence are also likely necessary to entice and maintain customers, probably more-so an an industry like insurance in which there aren't many tangible differentiators between competitors (see Coke vs Pepsi).

I don't see why insurance should be any different than any other industry regarding these issues and also the existence and legitimacy of profits (assuming they weren't derived from fraud).

It is different in this that you can't evaluate the price of the thing you buy. It's like contemporary art if you will, but art is more obviously a scam, while insurance have an aura of solidity and serious.

"fair insurance", "There should be not one penny left ", "Lottery pleasure is empty"... You are stating a lot of opinions but not many facts. Also, why isn't it "fair" for insurance companies to turn a profit while it's fine for other types of businesses?

You can have a look in a book titled Thinking Fast and Slow. It is strongly backed by multiple experiences. Profit: it is ok to make profit, but where ifs the limit? Would you agree that companies should not take advantage of children weaknesses to force sell them useless thing (a lot do)? Or mentally deficient ones? I wouldn't, because somehow we are not in a jungle. And the point is that humans are not grown ups in front of rare events. Everyone fears an earthquake during the week following an earthquake, and completely forget about it the next month.

Moreover the fact that insurance are highly regulated just makes my point. We are defenseless and the law puts some limits because a scam is too easy to build.

Humans are not homo economicus.

Sure and that's why they are easy to beat. How many industries would not even exist if human were rational? And again, the most rational ones have less loss aversion, take more pondered risks, invest in what has real value (education) and usually win the game in the end.

Insurance is about managing risk. I willingly and happily pay extra money spread over many small payments to eliminate the risk of a single rare event (e.g. A car accident that's my fault) from totally ruining my life.

The car accident is different, insurance is mandatory because you also risk other people's lives.

And I don't say you shouldn't pay for your own protection, I say you can't evaluate the risk correctly and are very likely to be ready to pay too much. Which is how insurance make money.