Insurers will encourage you to go to one of their “preferred” body shops because it saves them money. They have a contract with the body shop to pay a reduced labor rate and use generic or used body parts. Their body shops often cut corners just to turn a decent profit on a lot of insurance jobs.
But it is illegal for an insurance company (yours or another party’s) to require you to use any specific body shop! The legal term for an insurance company trying to force customers to specific body shops is called “steering”. It’s a huge issue in the auto insurance industry and only a handful of states have taken a firm stand against it, unfortunately.
If your car is paid off (no loan or lease), which I assume it is since you only have liability coverage, you are under no obligation to have it repaired. But you are owed for the damage because you have suffered a financial loss (loss in value to the vehicle or the cost of the repair).