What do you mean “by encouraging them to break their contractual relationship with their insurance company?” The presumed at-fault party’s insurance company is encouraging OP to utilize that contractual relationship with OP’s insurer…..albeit as a way for at-fault insurer to gain some advantage themselves and save money or hassle.

OP’s insurance being required to pay for the damage if prompted to by OP is tied to whether OP has collision coverage on the policy and not tied to whether other driver was at fault. They would pay it minus deductible and likely subrogate and try and get deductible back to OP.

As a personal injury attorney I disagree. We don’t have enough info to say it is worth it here.l especially if there is no bodily injury. There is nothing to suggest OP cannot reject the offer and demand the other driver insurance cover the damages as assessed by one of their accepted shops.