What this plugin misses is an adaptation of the registration form. This one still uses the username field. One could create a language override for the Username field, but the form still has 2 e-mail fields apart from that, resulting into 3 e-mail fields. Is there a way to have the registration field no longer show the username and maximum 2 e-mail fields?
I think I might revisit this extension to make it work more like you're describing.
I wrote something similar for another of my sites, and it handles usernames in the background. Every time someone registers, it creates a timestamp/ip-address based username for them, and when the registration is complete, it overwrites that username with something generic like "username1234" (where 1234 is the user ID). All of the username fields are hidden in that extension.
It was purpose written for a specific site, and I'm not sure it would work if I turned it into an extension.... I'll have to do some testing to see if I can turn that into a replacement for the Authentication - Email plugin or if I need to rewrite it to be more portable.
It would be very nice if you can get that done. On the other side it might be worthwhile to see how Joomla 4 treats user logins, maybe logging in by e-mail is available natively. My site runs 3 anyway, so I'm still rooting for an extension that hides the username field and fills it in automatically, as you describe,