Large public systems could do away with usernames if they assigned random, unique passwords to users out of a large keyspace. Usernames allow the keyspace to be

Large public systems could do away with usernames if they assigned random, unique passwords to users out of a large keyspace. Usernames allow the keyspace to be smaller and the keys have less entropy making user chosen keys possible. But then we add key minimum entropy requirements (sometimes using only length as a proxy) in order to combat the freedom given. Would be interesting to see whether username-less public systems suffer fewer intrusions. Some users would record their passwords in unsafe places, but they couldn't reuse a password eliminating password stuffing as an attack surface. Of course, if you allow password resets you'd still have a kind of "backup" username presuming a person's email address could be used for the reset. But that wouldn't significantly weaken the security, arguably.