Fiddling with the regular approval messages, I realized that there is a direct URL for the email me a login link:
/login-helper/send-login-mail?login=USER_EMAIL&destination_url=%2F
It seems to work fine, but you do need to prefill the email address and a destination.
Just for reference, you can pre-fill the password reset UI as well:
/password-reset?email=USER_EMAIL
I’ll mark this as the solution for future visitors!
Edit: I removed this as an answer, due to the issues highlighted in further posts.