Mobile phone authenticator apps are not that great anyway. They rely on the security of a phone that is connected to the net to keep the keys secure. If someone can get code running on the phone and exploit a privilege escalation vulnerability to get root, they can read the keys. A hardware token like a Yubikey or similar is probably more secure.
Another problem with the Google authenticator, at least for Google accounts, is that you have to add a phone number to your account to use it I think and, they then allow access by SMS, which is not so hard to circumvent.
If you're going to use Google Authenticator anyway, at least unlock your bootloader before you initialize the identities. You can't get to the key database without being rooted, even if you're doing a full backup with adb, and unlocking/relocking the bootloader wipes the device.