The only scenario wherein Craig Wright could actually be Satoshi is if Satoshi had burned his bridges -- in the process of abandoning the Satoshi identity, he had deleted all cryptographic evidence of the identity; all private keys and credentials for online accounts.
In that case, if he later (now) decided to assert the identity, then he might be tempted to cobble together fake proof to get his foot in the door far enough that he could begin to assert various forms of social proof with people that he had interacted with while using the Satoshi identity.
It's worth noting that the "scorched earth" elimination of the Satoshi identity is another possible theory as to why nobody has credibly claimed the identity -- because even the individual (or group) that assumed the identity no longer has the ability to cryptographically assert that identity.
People have suggested it's possible that the real Satoshi could have destroyed or not kept the original private key, or it belongs with one of the other, now deceased team members. Is there anything to gain by not admitting to it if this were the case?
Even if a message were to be signed with the original key, people suggest that it would still not be proof as it could have been stolen or extorted. It seems there is no way to prove anything at this point.
CSW's backing by "experts" suggests he is not an ordinary conman. Surprisingly, none of the experts have so far backed down on their support of CSW. Even GA's statement does not withdraw support, only expresses regret for posting support too soon. Also CSW's boldness to assume that the real Satoshi would not out him at this point makes it likely that he was involved with the Satoshi team and has some knowledge about his identity (deceased or otherwise). Or, he is gambling.
CSW's writing style in his blogs and elsewhere suggest it cannot be the same person as the author of the white paper, and yet I've heard another say that his academic papers contain a writing style close to the white paper (I have not verified myself).
All of this is obviously bizarre - the backdating of evidence, fraudulent cryptographic proof, etc. Either he severely underestimates his audience, or he wants to be be discredited.
There has been a suggestion that CSW wants to discredit himself to throw off extortionists (reported 6 months ago).
> Is there anything to gain by not admitting to it if this were the case?
Well, if CSW is Satoshi (I think it's pretty clear that he is not; this is more of a theoretical exercise) then his attempts to gain enough credibility to at least communicate with the people that can verify (through social proof) that he is Satoshi would not be helped by making a claim like that, since it is impossible to prove.
My problem is that I don't have a good idea of why CSW is going to this trouble to claim to be Satoshi; the idea that he's a regretful post-scorched-earth Satoshi at least would be a credible motive, even if extraordinarily unlikely. The only other options are that he hopes to gain something -- maybe obtain employment or loans on the back of the reputation of Satoshi.
The Satoshi coins can be publicly destroyed so they will never be spent. Transfer them all to an address like 1BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB. This is much more trustworthy than leaving it in doubt whether the coins can ever be recovered or not.
Well, in the "scorched earth" scenario, it would be more accurate to say that they "could have been" publicly destroyed. It's not clear to me why public destruction is necessarily a part of abandoning the identity. I don't understand what you mean by "trustworthy", though; if the identity is abandoned, then what's to trust?
In that case, if he later (now) decided to assert the identity, then he might be tempted to cobble together fake proof to get his foot in the door far enough that he could begin to assert various forms of social proof with people that he had interacted with while using the Satoshi identity.
It's worth noting that the "scorched earth" elimination of the Satoshi identity is another possible theory as to why nobody has credibly claimed the identity -- because even the individual (or group) that assumed the identity no longer has the ability to cryptographically assert that identity.