I've tested the login with my own Gotosocial instance (zlatko@social.zlatko.dev) as well as with my pixelfed.social account (zlatko@), I never got any DM with the Auth code.

Happy to help troubleshoot if I can (ee.g. I can look into the logs on the GTS instance).

Also it's not a big deal, if you're busy or don't want to deal with this now.