The only time I probably wouldn't go through insurance is for a slight ding that could be put right via PVR type repair. In fact I just did that with my new Sportage, reversing in work car park, in the dark, unfamiliar/longer vehicle, no parking sensor on the base model etc. etc. just clipping the side of a protruding wall with the tail-gate edge.
My fault, so as no other party involved phoned around a few local repair companies and eventually got it put right for just over £200. Quite a good job, certainly not evident to the casual glance. However, had another party been involved whether my fault or theirs, I would have gone through insurance regardless of who's fault.
Years ago I had a woman back into my Sierra in a car park when I was stationary and parked up. No witnesses and long before the era of dash-cams etc. but despite admitting liability at the scene, she recanted during the claim and it eventually went to court, where she didn't turn up and the claim went against her by default. Had I accepted a give me a quote and I'll settle in cash deal she would have denied it later, stiffing me with the £500 - £600 repair bill.