Contemporary insurance schemes usually rely on the insurer having some means (or some chance) of possibly recovering the sum, either by siccing their lawyers on the thief/thief's bank or getting the police to step in.

With bitcoin there's really no viable way to do this as there's no easy way for a sub-NSA-level actor to trace BTC transactions.

So I'm at a loss as to how such a system could be made to work.

I don't think this is true for many types of insurance. In many cases, insurance companies can "go after" a perpetrator, like in car accidents and credit card fraud situations, but what about home insurance that covers fire/flood/acts of God? What about life insurance or health insurance?

In these cases I am pretty sure the insurers make their living by shrewd actuarial rate-setting that tries to make their "average" policy profitable, and by having a large enough customer base that the "average" case is statistically likely.

The way to make BTC insurance work would be similar: calculate the costs to replace BTC, times the likelihood of needing to, and charge more than that.

What would stop someone sending them to another address which they own, and then claiming they were stolen?