The lack of key backup and restore is one big reason not to use Google's authenticator app. Other compatible apps are not so brain dead. I backup every time I add a new sign in.

If you don't have the ability to sign in from multiple devices and the ability to install access onto any new device, then you're doing it wrong.

Phones are highly portable devices subject to being stolen, damaged or just dying for no obvious reason --- so always be prepared. This is simply not possible with 2FA over SMS.