hi
keeping the car is not a disadvantage the insurer has decided the damage constitutes a total loss, it will now move to settle the claim asap, its likely a engineer assessment will be done via images of your vehicle that's why they want it moved to co part who will supply the images to your insurer ( not allowing removal may delay the claim also) so unless you intend to try and negotiate to retain salvage no advantage to keep it with you .
your input will be down to negotiating a fair "market value " to do this you need to prepare supporting evidence IF THE OFFER FROM THE INSURERS IS NOT ACCEPTABLE i say this because i have settled claim first time by a fair first offer !! , which includes cars of same , age, mileage , condition, spec, or as near as you can find, this incs all sale sites, any publication /on line sales ( not any part exchange deals) this could inc trade sources such as parkers.
your insurer will settle your claim and reclaim the expenditure from the third party insurers , subject to the t/p insurers accepting liability ( not the same as the driver saying it was my fault!)until its reclaimed its a claim on your policy.
not taking a courtesy car will not increase the valuation of your car.
the insurance is to put you back in the position as if the loss had not occurred.
happy to advise further
p.s to put this in context i have been a motor vehicle insurance engineer for 39 years .