Keep in mind that you can always buy the car back for its "salvage value" if they do total it. It'll have a salvage title, though, which will forever reduce its resale value and may make it hard to get collision or comprehensive insurance on.
'They' can do no such thing, because 'they' neither own the car nor have a contract with you, nor are you required to give them the car in order to be paid for damages. (Though what they'd prefer could well be different.) Probably doesn't apply to no-fault states, and certainly doesn't when _you_ claim against _your_ insurance, where a pre-existing contract applies. Just some more free lawyerly advice from an engineer. -- Jim