Generated username not sufficiently unique

If I create a registration form that has an email field but not a username field, and I allow login with email only, the account is created with a username generated from the email address. In fact, the username of the new account is the text of the email address up to the ‘@’ symbol. This is likely to not be unique. So, someone registering with [email protected], the account will fail to be created if there is an existing account created with email [email protected].

Should the username that is generated when no username field is present in the registration form be the entire email address? This would be ‘more unique’.