My understanding based on my own experiences, the person who's name is on the ownerahip doc owns the car until a settlement figure is mutually agreed.
Salvage co is not involved until the owner is paid out and the vehicle transferred to the insurer as the owner is just that -the owner of
the vehicle.
Joe public can buy back from the insurer in the same way a salvage company does.
The estimator who wrote off the vehicle will decide on the true value of the vehicle plus also it's salvage value. True value being the max they will pay out to the insured party (keeper) and salvage value being the figure they will sell it on to a salvage company (or indeed a private individual) for. Salvage yards will then sell on for as much (profit) as they can get.
Insurance companies and salvage companies have 'relationships' to maintain and subsiquently do not always want other people getting involved.
The insurance company simply looks at the value of the vehicle and the cost of the repair. if the cost of the repair is too high for the value of the vehicle then it's classed as a write off. the insurance company doesn't take into account whether the policy holder is at fault or who else is involved in deciding if it's a write off. they are secondary issues which solely relate to who eventually will pay for the cost of the write off.
the 'relationship' between the salvage company and the insurance company is simply monetary. there is no special relationship. the insurance company rarely sells the vehicle to the policy holder as it is too much hassle to value it. the salvage company does as it is their engineer that values it and assesses how much the repairs would cost once they receive the vehicle. they will, at times, say to the insurance company in their report that it is not an economic write off and should be repaired.
the salvage company (engineer) is involved with the vehicle and assesses/ inspects it which may occur at wherever the vehicle is taken after the incident. That may be the owner's drive (if it is mobile), the salvage company's property or wherever before the owner accepts the write off offer from his insurance company.
the owner of the vehicle does not have to accept it being classed a write off and may decide not to claim on their policy and accept the insurance company's offer. it may be that the vehicle is perfectly roadworthy and as safe as it was before the incident. the damage may simply be cosmetic but more costly than the vehicle is worth less the excess.
I know this because I work for an insurance company in this area.