Sure, but they can say, this is the shop we will pay ourselves, or you can go somewhere else and we'll cut you a check. If you need the repairs done faster than they cut checks and don't have cash on hand, it's a strong incentive.

Which is why people should have savings and learn some life skills like basic auto repair. Body work and paint I can't do myself, but that's mostly cosmetic (look at the number of beat shitboxes rolling around in the US). But you should be able to get a car into running order or acquire a rental vehicle while you wait for the check. Rental vehicles are usually at preferred rates when associated to an insurer and depending on the policy are covered.

Yep, the only item they can dictate is where the estimate is done.