Hello clankiller !
Thanks for reaching us today and sorry to hear about this experience with the plugin.
Based on your description, it is clear to me that things were working as expected before, so I would first want to ask you what has changed on the site since the last time emails were working? Were there any plugin or site/server configuration updates or changes?
At first glance it might be that something in the email setup might have changed, or maybe the SMTP is not configured properly or inactive. (these can be checked along with your hosting provider, as they can quickly determine if anything is wrong in your email service or SMTP configuration, and can usually guide you quickly towards any problems or solution)
Also, just so you know, backup codes can be regenerated at any time form your user profile, so no worries on that! More info here https://melapress.com/support/kb/wp-2fa-what-are-2fa-backup-codes/
In regards to your question if a plugin reinstall might solve this – it’s hard to say, since this seems like an email deliverability problem, but if you would like to start afresh and reconfigure the plugin just to try this out, you will first need to head to plugin Settings, and look after “delete data upon uninstall” option, then proceed with plugin removal and reinstallation once again.
(just so you know, by doing this, you will need to reconfigure plugin afresh, as well as all users will need to reconfigure 2FA)
Let me know how this goes and if there are new information in regards to this!