This decision belongs to you. The insurance company should not decide where you get your car fixed after an accident. Instead, you should make the decision that is in your best interest.
Some Things to Think About
Although you are not required to use a shop that is pre-approved by the insurance company, you may find that a pre-approved repair shop will have an easier time dealing with the insurance company on everyday issues that may arise, including cases in which the repairs exceed the estimate. Therefore, it may be less stressful for you to use one of these pre-approved repair shops - particularly if you are confident with the work that is done in such a shop.
That being said, and with the knowledge that the insurance company will only pay for repairs that it believes to be reasonable, you are free to have your vehicle repaired anywhere you are comfortable. Thus, if you do not believe that there are any pre-approved repair shops in your area that will treat you fairly then you might consider having a different repair shop take care of your car.
Car repairs can be expensive, inconvenient, and stressful and they may occur at a time when you have bigger concerns such as recovering from your car accident injuries. Accordingly, it is important to think about your options and to make the decision that is best for you.