Conservationists and archivists (sometimes) use gloves on cleaned samples, but not archaeologists. An archaeologist wearing gloves is almost certainly just cold.

There's no real reason to immediately protect the find from light handling when it's discovered. It's a dirty object that's still coated in whatever was floating around the environment for those thousands of years. The rock will be fine and probably handled multiple times before it gets back to an archive.

I think the only times I've worn gloves in the field were for a very rusty metal dagger and some especially sensitive human remains.