I don’t know directus but I have a guess that you need to configure an email server for it to be sent. You’d want to look in your app logs when you request the password reset to see if something comes up and directus docs as this doesn’t seems a Fly.io related issue.
I’m also trying to get this combo to work. But lubien is correct, the base image of Directus didn’t have sendmail (the default) set up when I tried earlier this week, so that email will fail. One thing you can try is not setting an ADMIN_EMAIL and ADMIN_PASSWORD and then looking at the deploy logs for that information. If it still doesn’t work you can probably assume the issue is with something else