The issue for me was that they wanted my Apple ID. Not the password, which I'm accustomed to typing and have got memorized. The ID itself, which is practically impossible to reset without concurrently changing the password.
Turns out I had two of 'em, resulting in a total of seven (really) auto-generated e-mails from apple.
:: sigh ::
:: closely related rant about Syllabic search edited out on the grounds of extreme incomprehensibility ::