If the question is: how do we prevent sale of stolen bikes, I’m not sure how that solves the problem.

Is there a possible way for FB and other similar places (offerup, pawn shops, etc.) to verify without some central registry?

Traditional retailers of used goods like pawn shops are usually subject to pretty stringent regulation - not necessarily to determine if goods are stolen, but to record identification of the seller to enable an investigation later. In some places this does include proactively sending a report of items to the police for checking against stolen reports. So there's a pretty clear precedent for regulation of this kind directed towards online marketplaces, but you can no doubt predict the questions around the difference between a pawn shop (being the seller) and Facebook (being not the seller but just a provider of listings).

bike index has an open API

And hurt everyone who is legit selling their own bike too.