We don't particularly care who has these items, but we want to remove incentives for randomly digging them up. The archeological value of documenting the context of an item is larger than any analysis of item itself, and once an item is removed to be sold, that contextual value is permanently destroyed.

Gaining an ability to track and possibly recover existing artifacts is nice to have, but it's actually harmful if it means creating a financial motivation for removing new artifacts and thus destroying the context for them; ideally, we want to create a situation where they get left in the ground because it's not worth for the looters to spend effort digging them up.