Yes, here in Washington (laws vary a lot across the country) Guitar Center has to abide by the same rules as pawnshops. That means photocopying a driver's license and taking a thumbprint. They also have video cameras over every point-of-sale terminal. Of course GC knew exactly who the sellers were.
However, burglary is a highly organized criminal enterprise. The person who sells the stolen property is usually at least two degrees removed from the person who committed the burglary. The robber will immediately transfer the goods to a "wholesaler", who then parts the loot out to specialists based on the type of item. They in turn hire a flunky to assume the only risky part of the process, selling the stolen items. This'll often be a drug addict, teenage runaway or homeless alcoholic desperate for cash, someone who does not even know the name of their employer.
Of course, a dedicated detective
could follow the chain back, but that would take a lot of time and effort. In my city, there are only two property crime investigators and they receive 10 to 20 new cases every day. The best they can do is patrol the pawnshops and try to match some serial numbers. They have about a 5% recovery rate.
Even if they find it, though, pawnshops (and music stores) are not obligated to give the stuff back to its rightful owners. Strange, I know. When I spotted my two stolen amplifiers at GC, I thought I could just walk out with them. Nope. Those amps are in storage somewhere. If I ever do get them back, I will have to pay for them.