Not OP, but customer identity is a component of my work. Ask users for a recovery email and/or phone number to bootstrap identity if sign in with goes sideways.

Definitely save the email in your database — even if it’s a private relay address. Also, send a welcome email right after signup, so users can see which email was used and ideally encourage them to update it to a regular one or add an alternative login method (like passwordless email sign-in or OAuth).

If we were starting over, we’d make that update flow more prominent from day one. Apple’s “Hide My Email” sounds harmless until it silently breaks everything later.