Except, of course, that if the insurance paid out on the stolen bike, it's now technically theirs.
I just did a bit of research (ie, I googled). Seems that the general policy is yes - if you claim, and the item is recovered, it's the insurance company's, but in practice they usually say "keep it" when you contact them to offer it back to them. I imagine it's not worth their bother trying to sell it on.
But I suspect this is one of those twisty little situations where, if you don't offer it back to them and that fact subsequently emerges, they might be quite sniffy about it - and defrauding an insurer is one of those things that can have quite a profound effect on your premiums later.
Since
Diamond is in one of those jobs where the appearance at least of probity is a very important thing (not to mention the fact that his recovery of the bike is now a matter of public record), in his shoes I'd be falling over myself to put the insurance company in the picture, all the while keeping my fingers firmly crossed that they'd just say "keep it".