It's just a way to help police recover stolen property.
The fear is that the government of Illinois may then have an easy way to confiscate that precious metal from private owners. Maybe Illinois residents have reason to fear their government. But residents of other states aren't quite that paranoid, at least of their state governments.
In fact, the state of Texas has had a similar law on the book for almost a decade. With the price of precious metals near their peaks there is a steady market. Transient precious metal buyers will set up shop in a hotel and advertise for people to come sell their jewelry. Seems like an easy way to unload stolen property.
House burglars will look for anything of value. But let's assume the burglar made off with a bag full of expensive jewelry. Unless he just likes to give jewelry away to his many girlfriends, gifts that carry a definite risk, he probably wants to turn that hot merchandise into cash. Along comes a precious metal buyer who may just want to melt down the gold or silver to sell in bulk. And it's a perfect crime.
The Texas law requires that dealers of crafted precious metal mail a description of the property along with the name of the seller to the chief of police or sheriff within 48 hours after the description is supplied to the dealer.
Yes, I suppose that does impinge on a person's privacy when the police have a description of jewelry that just changed hands. So it's a trade off. Sellers have to give up information. And police have a way to recover stolen property.
On a related note, way back in 2010 a guy was arrested for robbing a Cash America Pawn Shop in Lubbock, Texas, after he tried to sell jewelry he stole to a precious metal buyer in Florida.
Sometimes the good guys win.