Sorry, you're incorrect about repairing the car at any cost. They will work on the exact same basis as any insurance company, as the law works in the same way and they have to mitigate losses regardless. Remember, ultimately they have to go to the TP insurer claim the money back, and if they were extravagant then the insurer may well turn the claim down and take the process to court, and there's no guarantee the AMC will win. No, if a car is uneconomical to repair then that's it I'm afraid, using an AMC is not a guarantee that everything will be rosy. Worth remembering as well that AMCs are more expensive than insurers, so every time you use one and take a flashy hire car, what you actually do is push insurance costs up for every one. Not saying I wouldn't use ever use one, just throwing the info out there.
An insurance company cannot write your car off if you don't want them to: It's YOUR car. You can simply decline the claim, and then attempt to repair the car yourself and no CAT marker will ever be added to the data of the vehicle. Not always financially practical, I agree, but it's the truth.