The theft insurer has an incentive to make your car safer, and discourage theft of your car. For example, they might pay you to install a LoJack GPS tracker in your car and add a bumper sticker advertising the presence of the LoJack system. This is a real thing which real insurers do in the real world.
The theft insurer also has an incentive to recover your car if it's stolen: then they don't have to pay you back! Indeed the theft insurer even has an incentive to maintain a private police force which violently assaults criminals to take back your car. This idea is what's called a "private defence agency" (PDA) by anarcho-capitalists: a PDA is just a theft insurer which is a bit more active in discouraging theft. It is not a real thing (it is illegal), but the incentive towards it is real.
How much would the PDA spend on trying to recover your car through violence? The PDA would be willing to pay an amount up to your coverage. If they'd have to pay more than that to recover your car, they'd prefer to give up on recovery and instead just pay you the coverage amount.
So if the thief spends a little more than the coverage amount on stealing and concealing your car (assuming for simplicity that a dollar on theft and a dollar on recovery perfectly counter each other), then the thief will end up with the car. If the thief spends any less than the coverage amount, the car will get recovered.
Now: what if the thief goes to your PDA and offers to "bribe" the PDA to not protect, or try to recover, your car? The thief would have to provide a "bribe" which is a little more than the coverage amount, but then the PDA would simply give the thief the car and pay out the coverage to you. The thief wouldn't offer substantially more than the coverage amount, because they could always just steal the car instead for just a little more than the coverage amount. This avoids the value-destroying conflict, so the thief and PDA are quite happy to do this.
Anarcho-capitalists are now frothing at the mouth to tell me how stupid I am and how this wouldn't be possible and how the reputation of the PDA would be ruined. But wait: is this "bribe" situation actually a problem? No!
There's a popular activity where someone gives you money and you give them your car: it's called "selling your car". The only difference here is that it's not optional: you must sell if someone comes to you with an amount exceeding your insurance coverage, since they could just take it otherwise. But that's fine: you have some dollar value for your car, just like every other possession, and if someone offers you more than that amount, you'd be willing to sell the car. So you just need to make sure your insurance coverage exceeds the dollar value you have for the car, and then you're also happy to do this transaction.
And a "thief" would only offer this size of "bribe" if they valued your car more than your coverage amount: in other words, if they valued your car more than you do. So selling your car to this "thief" is actually socially optimal and efficient in the Coasean sense.
We could imagine that this process would be systematized: the PDA would publicly list all the cars it covers and the amount of coverage, which is effectively a sale price. At any time, a buyer would be able to come along and buy a car by paying the coverage amount to its current owner, plus some small transaction fee which goes to the PDA.
So in the world of PDAs, "private property" would be substantially weaker than in our world. You only control some property as long as:
This actually lines up neatly with a very real proposed reform: Harberger taxation. The insurance premiums paid to the PDA are equivalent to the Harberger tax, and your coverage amount is equivalent to your self-assessed value. There are many benefits to a Harberger tax which I won't explain here; it has a lot of theoretical economic justification. I think Harberger taxation is actually a great policy for the real world!
Why have anarcho-capitalists never embraced the fact that their proposed society would do away with private property? I think they're all too distracted being edgy and right-wing. Ancaps flirt with ideas like slavery and private nuclear bombs, but they turn up their nose at a little abolition of private property, even when that's a well-justified and natural economic result!