> Any who is going to in charge of reading that proof of identity and moving the coins? A trusted centralized party?

Basically you'd have to relax the trust/decentralization guarantees, but you don't have to relax them all the way. Most likely a consortium of trusted actors (Blockstream, major miners, major exchanges, bitcoin-adjacent companies,...). Or something like a consensus mechanism with aligned incentives a la Kleros. I think "we" could come up with "something", even if it is not perfect, because the value of Bitcoin is ultimately in the community of people who use Bitcoin, not just the protocol.

"Hard-fork" might not be the right way to see this. It's more like starting a completely new protocol where people who held Bitcoin at a certain snapshot can redeem a one-time airdrop equivalent to the value they held, provided they can prove ownership. As that protocol's value overtakes the value of the original Bitcoin chain (which will eventually be completely dead), we can all agree to call it Bitcoin.