What exactly do you prove once you mine a genesis block? That you can achieve it with difficulty=1? As I said, you'd do that after to maintain the longest chain. What difference would it make if Satoshi had picked the first hash with nonce=1? There's nothing to be proved in the genesis block.
Probably nothing. I'm not sure whether Core skips the checks for a valid block when it comes to the genesis block. I'm not too familiar with the codes for Bitcoin Core but I'm pretty sure I've read somewhere that the genesis block still has to have sufficient PoW and was validated as per normal for at least in the first few release. Perhaps someone can add on to this.It really does skip the validation checks for genesis block, at least for mainnet and signet, and I'm sure that's how testnet works too. mainnet signet
Why do I have to mine the genesis block? A forum member had posted an announcement thread of his project ([ANN] Genesis Block Generator), but it is described that you'll have to mine the genesis block.
You only have to do that if your blockchain is empty (like his), like Satoshi did in the early days.
