Well, here's something insurance companies usually won't tell you. If you file a claim and 'just leave it to the insurance companies to fight for you', even if the accident is not your fault, it'll still count against YOUR record, and there's a very good chance your premiums will be higher either immediately, or upon renewal.
Insurance companies are in the business of collecting money, not paying out money. It doesn't matter if they collect thousands of dollars in premiums over many, many years. Irrelevant. If they have to pay out anything, in any manner, on your behalf, they'll look for a way to recoup that payout, many times over.
The 'goal' is to collect $$$ from the insured for decades, and never spend anything beyond some paperwork