With the aircraft no doubt being assessed as a total write off who decides on the outcome?
The writer of the "write off" decides the outcome. That's the nature of such things. It is, of course, negotiable between the insurer and the insured.
In some cases the wreckage in a hull insurance case has a positive value. In other cases the wreckage has a negative residual value due to the cost of recovery. That balance is what determines who gets to 'keep the kids'.
If some idiot taxys a 747's wing into an airport building, the aircraft may be beyond economical repair, but the wreckage of the airframe may contain many hundreds of thousands of dollars worth of salvageable parts. That value is taken into account in the final settlement.
In the case of Crash Tracey's most recent cockup, it's entirely likely that the close-out of the "write off" has resulted in her side getting to keep the debris.