Google authenticator blank after backup restore
You can't restore the Google authenticator, you must reinstall it.

If Google Authenticator is anything like Microsoft's authenticator, it is tied to the device, not just the account.  This is how Google knows it's actually you because you are using a device that you have in your posssession (it wouldn't be secure if you could use it on any device anywhere).

When you restored, you restored the Authenticator tied to your old device and since the devices are technically different, nothing comes up.  This is expected behavior as the app intentionally doesn’t include data in an encrypted backup to allow your tokens to transfer with the rest of the device backup like Apple keychains and health data and other iOS saved passwords and settings.