eBay has an anti-counterfeit program but does have some theft issues...but it's MUCH harder to move volume on eBay so it's not even close to Amazon's issues (not to mention some stolen goods are even housed and fulfilled by Amazon itself through their FBA program!) Facebook marketplace is also flooded with stolen goods, but that's largely with the not so smooth criminal minds moving low volume goods. Amazon has actually started to remove abilities to get ahold of support to report problems, there was a thread about it here recently.

The problem isn't with allowing third party sellers, it's with allowing unvetted third party sellers.