I Don't recommend to use 2FA google authenticator as its's almost impossible to recover your account once you lost your codes. I've lost my device in which my codes were saved
That is your fault, smart boy didn't save the backup code on the same phone. Backup code is just number, you can easily write it on paper and keep it far away from your phone. Or if you lazy to write the code, first time set up Google Authenticator, simply make a screenshot of the barcode and print it.