If you were Satoshi, what would be the benefit of claiming that in public (think it through at least twice before replying)? When someone gives a reasonable answer to that question it might be worth at least put their claims to test, which is pretty simple, just move Satoshi's stash of BTC, end of story.
If by “Satoshi’s stash of BTC”, you mean some huge amount, then you are referring to the so-called “Patoshi” coins. Nobody knows if those are Satoshi’s; I tend to think not. (Why would Satoshi himself use mining software that behaved differently than the Bitcoin software he published?) It can’t be proved either way, unless Satoshi or Patoshi steps forward.
The threshold test for anyone who claims to be Satoshi is
authentication with a digital signature from one of Satoshi’s known keys: His PGP key, or the key for one of the Bitcoin addresses definitively known to be Satoshi’s. Not moving coins that may or may not be Satoshi’s, and that may coincidentally move while some random scammer claims they did it.
By “authentication”, I mean signing a message with substance similar to this:
2022-06-26: I, [name of claimant], am Satoshi Nakamoto.
I don’t ask someone’s motives before
demanding that. If the claimant refuses (as Craig Wright has), then his motives are obvious: He is an imposter.
To illustrate the dangers of relying on faulty assumptions about the ownership of early coins:
[...] under oath Craig wright provided a list of the bitcoin holdings he claims he mined as satoshi... thousands of early addresses. ... and as soon as the list was published the owner(s) of 145 of the addresses, controlling 7250 BTC,
signed a message saying those addresses didn't belong to Wright and that Wright was a fraud.
IIRC, those coins have not been moved. Their owner(s) evidently follow the news, and still have the keys.
The content of the signed message, which you should verify for yourself (see above link):
"Craig Steven Wright is a liar and a fraud. He doesn't have the keys used to sign this message.
The Lightning Network is a significant achievement. However, we need to continue work on improving on-chain capacity.
Unfortunately, the solution is not to just change a constant in the code or to allow powerful participants to force out others.
We are all Satoshi"