Should an African-American individual petition the US government to be resettled to Africa and compensated for the trouble, would the US government agree to pay? I don't think so. The US government is responsible for the enduring consequences of segregation, which ended less than 50 years ago in most of the country and is a separate issue from slavery, albeit strongly related. In the same way, the current UK government is responsible for preserving the artifacts it inherited so long ago and (theoretically) work towards a world where the Greek people will never be forced to sell their belongings because of economic hardship.
> The UK government should at least attempt to return the relics
Why this obsession with the British Museum, when the Louvre and other institutions are as full of less-documented loot as the BM? Are we holding the British responsible for keeping better records than others? There are tons of US and European museums with similarly-acquired catalog that are never really criticized for it. The Nike of Samothrace is so much more beautiful and powerful than anything the BM holds, but its location has never been much criticized. Could that be because modern Greek governments have had better relationships with France than with the UK? It's all just foreign policy by other means, usually by people who won't care about these relics the minute they've had their way.
I personally think the best location for artifacts is among people who will cherish them and help the world see them and understand them. As an Italian, I think half of the relics we currently mis-preserve should probably be sold to people who would do a better job. The recently-located Imperial Port of Rome will likely never be fully excavated because we wouldn't have any money to preserve it as it would deserve, and the same happens every day for magnificent stuff all over Rome. I think that's a much more pressing issue than bickering about stuff we've already studied to death.