The care of the artifact must be weighed against those who seek to claim it.
Even items freely given have had attempts to take them back. I also am greatly concerned with antiquities returning to areas which have experience real strife if not outright wars. Especially the rash of religion based conflicts even between sects of the same. We have seen the destruction visited upon irreplaceable sites and this must be a consideration before any object is sent to a claimant.
I'm curious why this is a part of your argument. If I spend a year staking out your house, and spend considerable effort to break in and steal your stuff... does that make it any more legitimate?
> they would probably be tucked away far from public view
IIRC for most museums the vast majority of items aren't on display either because they're not suitable for public display or because there simply isn't enough space to display them.
Perhaps what you fail to realise is that a great many treasures have been looted and sold into private collections over the years. Looted and sold by locals, to anyone who can pay. So there is a good chance that anything that the British Museum received as loot would have been someone else's loot otherwise.