In America, in many/most states that aren't "no fault", you're require to buy "uninsured motorist" insurance so if you are in a collision caused by another driver and the other person does not have insurance (because they're breaking the law) or their insurance is insufficient to cover your damages, that pays your damages up to the limit. Also many collisions result from somebody breaking traffic laws. (Failure to stop, failure to yield, making an illegal turn, speeding, etc.) Their insurance still pays.
You can set up a system to the insurance company pays up to the limit of coverage, and has the right to sue the insured to recover what they can if they are found to be at fault in a shooting, or where the insurance company pays whatever awarded damage is left to be paid after the shooter (or their estate) is tapped out.
In Austria (and much of Europe) all licensed insurance car providers must pay into a pool that is used to pay claims against uninsured motorists, up to the minimum coverage every driver is legally required to obtain. At the time of this writing this is 8 million EUR. Additionally, insurance is tied to the
vehicle, not to the driver, and the commercial risk of the driver not upholding their insurance contract lies with the insurance company. For example, if you have an insurance contract that says "no drivers under 25", and you let your 18 year old kid drive who then proceeds to crash into another car, your insurance company must first pay the claimant, and then may recoup the costs of your breach of contract from you. Note how the claimant is completely insulated from the consequences of a breach of contract on part of the insurance holder.
This creates a strong commercial motive for the insurance company not to underwrite badly scoped out limits on the insurance, because they carry the risk of the insured person not holding up their side of the bargain.
License plates are not issued by our DMV, but instead by insurance companies in name of the state. From the point of issuance of a plate the issuing insurance company is liable for damages caused by the plated car -- even if the insurance holder stops paying the insurance premium. In this case the only way the insurance company can escape strict liability is by reporting this fact, and the fact that they are withdrawing the insurance to police.
This creates a strong commercial motive for the insurance company to report persons who do not meet their legal requirements to properly insure their motor vehicle to the police.
But even after the police report, the insurance company is still indirectly liable through the 'uninsured motorists' funds they are legally required to contribute to.
This creates a strong commercial motive for the insurance company to exert pressure on the police that 'reported as uninsured' plates are collected, and that unplated cars on the roads are pursued and impounded.
Note that all liability incurred by the insurance company also exists against illegal acts: If you rob a bank and crash the getaway vehicle, the car's insurance still needs to pay for the victims of the crash, even though it happened in pursuit of a blatantly illegal and indeed criminal act.