I think even the proof that one is/was the owner of those private keys is not enough anymore. The more time passes, the higher the chances are for someone's privkey to collide with an already used one. While this is not a very likely scenario and the chances of colliding specifically with the genesis/Satoshi's privkeys are about zero, it's not practically impossible especially as tech evolves.. So if someone "finds out" 50 years from now who owned those private keys, chances are this is just a random coincidence and collision that proves nothing.
Use quantum computers to crack the ECDSA keys for those. Collision is quite a lot harder.
Best case would be to publicly post the private keys of Satoshi's addresses. That way, these frauds wouldn't manipulate a part of their supporters.
That would never happen. Satoshi probably won't appear again nor touch those coins.
True. As time goes by, these private keys have more chances of being found by someone else. When Bitcoin will have changed to quantum safe algorithms, someone may pretend of being Satoshi by searching the, insecure by that time, 1 — 2^160 range. In other words, in the far future, there won't be any ways to prove you're Satoshi.
Speed up for symmetric algorithms are far smaller and probably wouldn't be sufficient to find specific keys. It's not like there is a huge portion of known Satoshi keys.
I think you meant using QC to directly crack the public keys.