I had this same conversation in June with my insurer.
It boils down to regulation of costs -
If YOUR insurer repairs your car , they know the costs as per their network partners, and can budget nicely.
Take this a step further, if YOUR insurer repairs THEIR car as well, your insurer then gets to budget their repair costs as well. They will then ADD A REPAIR MARK UP and thats what they invoice to the other insurance company.
Now, look at this the other way around - Their company will offer to repair your car as well because it will be cheaper than them getting stung for your own insurance company repair bill plus a mark up.
Or, to put it another way, they can make a margin by billing the higher rate to your insurer instead of your insurer billing them (and your company making the margin)...
In fact my insurer virtually told me off over the phone for NOT telling the 3rd party in my bump to contact them instead of their own insurer.
So, regardless of fault, the insurer wants to do the repairs and bill to someone else.
Sounds good for them, but is it good for you?
ummm - not really - I dont see they'd be inclined to pass that saving / margin onto you once it comes to renewal time