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In Satoshi’s case, I think the proof of his identity would require more evidence:  It would be necessary but not sufficient that he provide a cryptographically signed proof.  It is unlikely, but possible that someone may have stolen Satoshi’s private keys.

The signed statement is the baseline, the first evidence to demand; without that, yes, “everything else bunkum, hot air, waffle”.  It would then be necessary to evaluate whether the claimant had the same level of cryptographic knowledge as Satoshi did, the same writing style, etc.  Given the amount of money that a scammer could steal by stealing Satoshi’s identity, we would need to be sure!

Craig Wright has not even provided the most basic evidence.  Of course, that’s why everybody who is knowledgeable about crypto immediately rejects his claims.  The danger is that he can fool newbies, and he can destabilize the market by making his claims to people who never used Bitcoin at all, and he can do other damage by telling his lies to the public.

That’s why the public must be told that Craig Wright is an identity thief who has stolen the identity of Bitcoin’s founder.  Words are powerful.  People know what “identity theft” is, so they will immediately get the right idea—instead of falling down into a complicated crypto discussion involving topics that are unfamiliar to them.