not_a_charger knows his way around these things and I'd go with what he says.

I'm not going to suggest that what follows is ethical or even legal: if it was me, I would have had it towed to my driveway rather than the local tow yard. When the hassle with the insurance company is concluded, they can come collect it. Maybe some merit in the idea in terms of valuable stuff disappearing while in the impound lot but my thought is entirely about some parts disappearing while it's parked outside my garage.

I really doubt they'd notice or care about stuff like rims, trim pieces, small interior bits. They'd probably notice and likely object to it missing bigger key components affecting the salvage value - front clip, motor / trans. Might be an opportunity to negotiate with your insurance company but really no clue if they'd be willing to play along. Reduce the agreed value pay out by the fair market salvage value of anything significant you remove. Could well be a better deal than buying back the wreck.