Basically, as you are presumably already aware, insurance companies are basically crooks who rip each other off with absurdly inflated hire costs, etc. And pass on the costs to their customers.
Your insurance company do not work for you. They work for themselves. Unless you are impecunious and the liable party's insurers are being difficult, going through your own insurers only benefits them. Potentially, you could be stuck with the crooked hire car costs if the other party's insurers refuse to pay - although I would love to see an insurer try to sue their own client for their inflated costs in such circumstances.
The liable party's insurers have a vested interest in controlling the costs and keeping you on-side. Your own insurer has a vested interest in using you as a patsy to rip off the liable party's insurer, and potentially throwing you under the bus for their actions.
edit: Some years ago, a driver insured with LV went into the back of my 11 year old car at a pedestrian crossing. Within an hour, LV were on the phone to me offering to sort everything out (although they stopped short of offering me a kickback). They asked what my requirements for a hire car were - and as I only needed to commute 15 miles each way, I said that anything would do. After all, they didn't make bad cars any more, and a new car would be far nicer than my old car. Or so I thought. They arranged for me to collect a 1.4i Corsa from a national car hire place 200 yards from my house. I did not understand the term "malicious compliance" until I drove that unmitigated piece of s**t. Arguably more Vauxhall's fault than LV's, but a cautionary tale nonetheless.
