Video is powerful evidence, but the insurance company has a point: they need an official document proving fault, not just a recording. The thing is, traffic police don't always write up incident reports anymore - they only do it if someone was injured or there's a violation that carries a penalty. If the other driver just denies responsibility, the officer can refuse to file a report, and then you're left with civil court.
Here's what will actually help: sue the at-fault driver for damages. The video will be your main evidence - courts accept it gladly because it's objective. At the same time, file a claim with your insurance company, attaching both the video and the court filing. A lot of insurers pay up quickly once they see you're ready to go to court. And here's a pro tip: if the video clearly shows the other driver's license plate and vehicle condition, try contacting their insurance directly - sometimes they'll sort it out faster than your company, especially if the damage isn't that big.
Also check if you have witnesses from the accident scene - their testimony combined with the video really strengthens your case. And save the video file in multiple places, ideally with metadata intact, to prove it's authentic.