Ownership of recovered insured property

A question inspired by the current Titanic thread and my watching How To Steal a Million this week.

Let’s say Mr Smith owns a valuable and unique artpiece. He has it insured. It’s lost or stolen - we’ll assume it’s a normal loss and Mr Smith was in no way responsible. The insurer pays Mr Smith.

Ten years later the artpiece is recovered. Based on what was said in the Titanic thread, the insurer now owns the artpiece. Is that correct?

Does Mr Smith have any claim to recovering the artpiece? Can he return the insurance payout and take back the artpiece?

Basically, it depends on how the policy was written.

Mr. Smith owned the property, but in effect he sold it to the insurance company when he accepted their payout. The policy may give him the right to buy it back by returning the payout. But the insurance company might ask for 10 years interest on the payout, though.

And the way the policy is written may vary for different types of goods.
A policy (or rider) for artworks (which often increase substantially in value) may be different from one for normal household goods (which the insurance company would have to sell, and probably get little for them – how much would they get for your 10 year old VHS tape player?).