It’s happened to me before. I never make a claim through the my insurer now. I notify but that’s it.

The other insurer might decline coverage as the other driver didn’t comply with the terms of the policy. Since the driver accepted fault, you can go directly to that driver for reimbursement. If both the insurer and the driver refuse to pay, report the diver to the state dmv for not having insurance. Then sue the driver and insurer. Your insurer might pay for your legal fees.