It seems pretty clear that they do have insurance for accidental damage, and that's exactly where they're trying to hide the fact that they don't have insurance

It seems pretty clear that they do have insurance for accidental damage, and that's exactly where they're trying to hide the fact that they don't have insurance against fraudulent renters. The risk profile for fraud is very different from accidents, which is why it's harder to insure against fraud, but also why you really need that insurance (or a deposit, or whatever).