Just because you share the same geographical location as an ancient civilization does not mean you have any relation or claim to them.

Actually that's generally how national sovereignty works, unless you do as the British did and loot artifacts from places you've violently conquered, and not even the Brits do that anymore.

For example Italy cannot just claim every Roman relic for themselves because ancient Romans and modern Italians are two separate peoples.

Italy can and generally does claim every Roman artifact found in Italy. Rome however covered far more than Italy, in fact Roman artifacts can be found in the UK and when they are guess where they tend to wind up? Under the control of British museums.

And the idea that Romans and Italians are completely unrelated is insane. Rome has been continually inhabited for almost 3000 years and I guarantee you that modern Romans consider themselves part of that direct lineage. Italian is also the closest modern language to Latin and that's not just a wild coincidence.

If the UK is making an argument that the artifacts shouldn't be returned because of the inherent risks then they should have no problem returning all the artifacts to countries like Greece or China. Since they generally refuse to return artifacts regardless of circumstances it's difficult to argue it's to protect them, it's clearly far more about possessing them.