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.
I wonder, might one instead make an argument that it's not currently economic-technologically feasible to avoid artifact surface information destruction and contamination, at any stage of the dig-to-archival-conservation pipeline, nor to extract whatever information might be there, outside of very rare cases? Rather than there's no information there to lose?
Perhaps it becomes possible-and-affordable at some point, to frequently check, say, "oh neat, here are some 10 um notches on the surface, with some nice yeast, and some pollen, and oh, maybe that's a bit of calcified biofilm for nanopore DNA?"? Does automated lab tech progress, someday make "brushed, handled, dusted, washed, and shelf in the basement" start sounding like 19thC "we accessed the site using dynamite"?
Moreover, just being in the general vicinity is enough for us to leave behind contamination, since humans (and all other animals) constantly shed detectable eDNA into the environment. And even if they wanted to deal with this complexity, the budget for an entire project's archaeological testing is often less than my coworkers' monthly token budgets. You might get less than a dozen radioisotope, TL, or dendro tests for the entire thing, let alone anything more expensive.