The only legit and practical reason on most web sites is to be able to restore access if you forgot the password.
So just near the password field we can have a "Restore options" section, with email, phone number, etc fields; each having a Test button.
If user have entered email address he can use the Test button and receive a confirmation email, with a link. When user clicks the link, we mark the email as confirmed in DB.
Also, as long as the user does not have a [tested] restore option, we can show him some indicator, like a small red circle on his profile symbol with a tooltip "no restore option configured".
Doing the test email in reverse for me personally would be annoying, because I do not have and do not want to have a configured email client.