Exactly, why would you sing a message saying that you're logging on this or that website, for everyone to see, when you can just tell the website in question and no one else?
You don’t need to post a message on the chain to ‘login’ to a web3 site. But if you perform a transaction such as changing the state in a smart contract or withdrawing tokens based on your address and message meeting some ZK-SNARK verifiable condition, that is when you might need to post a message to the chain.