I don't think you completely understand the concept behind the Google Authenticator App, i.e. the standard it implements.
Which keys is it supposed to backup?
Everything else you said is sadly true.
Which keys is it supposed to backup?
Everything else you said is sadly true.
Under the hood Google Authenticator uses keys to generate the codes you see on screen and these keys are not backed up.
It’s a difficult decision of course. If you back them up in iCloud Apple and people who hack your Apple account have access. If you don’t the keys are lost if the device breaks or is lost and you need a workaround.
Edit: oh you wanted to restore on the same device. Well that might work but it doesn’t help when migrating or if your phone is not available.