How to get the at-fault driver insurer to pay for car damages when the at-fault driver did not report an accident?
When the at-fault driver's insurer does not voluntarily offer you compensation for a car accident, you can sue the at-fault driver for the costs, damage, and/or injuries you suffered. If you win, the driver and/or his insurer will have to pay you.
There is tendency to over-rely on the other driver's insurance to pay for your car damages or other costs you incur, when someone else is at-fault in a car accident. If the other driver's insurer does not voluntarily ante up, how we can make their insurer pay? The answer is to sue but not the insurer--you sue the at-fault driver personally.
The at-fault driver's insurer is not like your own automobile or homeowner's/renter's insurance. The at-fault driver's insurer has no obligations or responsibilities to you. That's because you are not their customer or client. You are not the person paying for this insurance policy. You are not the one with whom they have a contract (since insurance policies are, at base, nothing more or less than contracts).
The at-fault driver's insurer is their insurer. The insurer's obligation, under the at-fault driver’s insurance policy, is to defend (such as in court) the at-fault driver, and/or to pay any amounts he or she is found to owe others due to an accident. Sometimes--maybe even often--the at-fault driver's insurer will choose to offer compensation (a settlement) without a lawsuit, but that's because in their opinion, the other driver is likely to sue and likely to win. Therefore, the logic goes, it makes sense to voluntarily offer the other driver something, rather than go to the expense of defending a lawsuit but likely losing, in which event they will have to pay for their insured anyway.
However, such a settlement is purely voluntary. They may choose to offer compensation but cannot be compelled to offer a settlement. And sometimes they just won't. Possibly they think that their driver is not in fact at-fault, or they think you lack evidence to prove your case, or that the damage you are claiming was not in fact caused by the accident. Whatever the reason, if the insurance company doesn’t want to offer you a settlement, they don't have to.
An at-fault driver--that is, someone who was driving negligently, or carelessly--is liable for the damage, costs, and injuries he or she causes. (The corollary is that a driver who was not at-fault in causing an accident is not liable or responsible for the consequences.) Therefore, if you sue the at-fault driver and prove in court that he or she was at fault in causing your damage or other losses, you can get a court judgment (an order) requiring the at-fault driver to pay. At that point, if the at-fault driver had insurance, the insurer should step in and pay for him or her, but they do so because of their obligation to the at-fault driver, not because of any obligation to you. And if for some reason the insurer does not pay (e.g., maybe the at-fault driver let the policy lapse), the at-fault driver will still be obligated to pay you.