Claiming that usernames cannot be a source of entropy/security needs foundation.
Back when the whole concept of authentication was new (UNIX) that was true because usernames were quite literally public information, you could see them via a directory listing. With early email (SMTP) that remained true but worse via public directories listings across-computer.
However in this context there's nothing inherent about a username that allows us to ignore its security characteristics. Unless the argument is "over the shoulder" leakage? Which I'd argue itself doesn't have a strong foundation.
Both obscure usernames and obscure passwords can contribute to the overall strength of a system. A system that allows the user to set their own password may gain particularly from pre-selected randomized usernames, as users have proved untrustworthy in the past when picking passwords (e.g. reuse, patterns, common words, etc).
As an aside, scrapping usernames and only having a password isn't inherently problematic, except two users with the same password may clash, and a password recovery scheme may be more difficult to develop. That's essentially what authentication tokens are.
> Generate a random unique password for your user and don’t have a username at all.
Because having an unknown username with an unknown password increases the difficulty of compromise via improved entropy.