1. This has nothing to do with the government. The timber is now the property of the insurers, as the vessel has been deemed to be a total constructive loss, as would your car be had that been stolen / lost and recovered. Would you be happy for insurance premiums to rocket is no lost goods were recovered?
2. The contractors have been appointed to collect the timber for one simple reason - to ask the public to do it would be dangerous, and few people are equipped for handling large baulks of timber either at sea or on the coastline.
3. It is, I suspect, highly unlikely that you would be asked to return, or probably pay for, any timber you recovered. It would have little or no economic value for resale. I doubt whether the tax on the miniscule amount someone could recover would be an issue.
4. Ships are rarely insured with a single insurer. There will be a lead insurer, with the rest spread across Lloyds open market or whichever country the vessel is insured with. Compared to the vessel loss, in this case the cargo loss will be a tiny percentage. The rescue tug operation won't have been cheap either.... This will affect the owners next insurance bill, and shipping insurance in general - not your car policy (or if it does, by about 1p).
They would rather the stuff rot or be swept around the seas
And damage / sink other vessels? B*llocks.
I dislike most governments as much as anyone else - and trust me, the French one ain't much better - but this is not about government. It is about safely recovering dangerous objects at sea, and ensuring that further vessels are not injured by it.