There's plenty of evidence that LLMs lie, cheat, and steal. Corporations are known for having all of the benefits of personhood with none of the responsibility. As more people are harmed through interactions with these non-human entities, insurers will start looking to those accountable and they will extract their pound of flesh.
Insurers are corporations. What makes you think they just won't pay out or will cease being useful as anything but value extractors?
Honestly, the fetishization of "Insurance will save us" needs to die. The risk doesn't go away.