Mandatory car insurance, in theory, moves money from drivers (proportional to the risk they represent) to victims (proportional to the damages caused), plus a cut for the insurance company.
I'm missing something in your flow of ideas. Is it a welfare benefit because that idealized theory of insurance is itself a welfare benefit or because when insurance is mandated it doesn't live up to expectations?
The insurance is mandated, and the focus is more on guaranteed benefits to the victims than on fairness for the people paying them.
If you can't afford to pay for the damages you caused, that's your problem. You took the risk, and now you face the consequences. There is no reason to make the insurance mandatory to prevent that. But if the victim doesn't get any compensation because you can't afford to pay, that's a public problem. Now there is a reason to make the insurance mandatory, and it's particularly important that the riskiest drivers have insurance. If you make it too expensive for them, they may choose to drive without insurance. Which is exactly what you wanted to avoid in the first place.
I'm missing something in your flow of ideas. Is it a welfare benefit because that idealized theory of insurance is itself a welfare benefit or because when insurance is mandated it doesn't live up to expectations?