Yes, the government does shred old money and replace it with new money.

It would be great if you could just sell your old coins and buy new ones, but what if nobody is buying?  Even if someone does buy your coins, then they are the ones left with worthless coins if the network dies.

So when a coin moves to a new network, how do you think people should transfer their old coins to the new coin in a way that doesn't allow people to hold the old coin for free?

You can call it taking someone's word, but all transactions are publicly available on a decentralized blockchain.  No need to take anyone's word.