I don't work in the insurance business (heaven forfend) but have investigated a similar situation. Pet insurance doesn't work like car insurance, there's no 'knock for knock' and they insure for your dog (and therefore you) only, not for the 'benefit' of third parties.
For example if two dogs have a scrap, yours is insured and the other isn't, your insurance pays only for your vet bills, not the other party's. They would have to claim off their own and if they're not insured, that's their lookout. You can't pay the other party's vet bill and claim it back off your insurance company, so if you do pay them, it's out of your pocket.
Very often there's no clear culprit (or no-one admitting to it), there are no witnesses (or impartial ones) and given those circumstances it's no wonder the police don't want to get involved.
Where more serious public liability cases are concerned, obviously the insurance company want unequivocal proof - ie by a court judgement - that your dog and therefore you ARE liable and they will not pay out until liability is proven. It's one of very few cases where you have to be 'guilty' to win.
If you are NOT proven liable, then they won't pay out. But if you are proven not liable, then you aren't liable and therefore won't have to pay either...
It is quite logical really, just don't expect to say 'It's a fair cop', fill in a couple of forms and be done with it.