The best insurer is the one with the most data and ability to analyse it. Nobody can beat the self driving car operator, they need that information anyhow to improve the capabilities of the car.

Interest alignment is also with the self driving company, they have any incentive to improve safety of the car. They are simply liable so they act as an insurer and just add that to the cost of the self driving module.

For the buyer this should remove most need of insurance. Even if there is some "manual mode" the obvious provider would also be the self driving operator. They could track driving behaviour and adjust the rates, or offer some limitations for accident prevention.

So the insurance itself is still there, but it will be less important and shift to the self driving operators. Even if you own a car, you would pay some fee for the self driving module for updates and liability in one package.

I worked for insurance companies. Car insurance is a huge business, but not a very profitable or even sold at a loss. It is often used to get new customers to offer them other more lucrative products. That going away is still a huge shift and those companies are very old and slow.