I would like the user to be able to register/log in only via email
Don’t.
You’re going to get angry requests from your users over privacy concerns!
That’s because all WordPress users have a PUBLIC URL constructed from their username, like https://example.com/auhtor/username
If they ONLY register with their email address, then that email address becomes their username and part of their PUBLIC author URL (with the @ committed and the dot . replaced with a dash -).
User john@example.com’s public URL becomes:
https://example.com/author/johnexample-com
And many, perhaps most, of your users aren’t going to like that for obvious privacy reasons.
Here’s one of MANY related support requests from WordPress site owners I’ve seen here: https://wordpress.org/support/topic/email-showing-in-author-url/
Do you still consider this a smart move?
