The biggest problem with Google Authenticathor is that you will need to manually back up every account in another device, or save the keys offline (manually as well).
If you do not save your 2FA in one device, than save on another, for every website, you will be depending 100% on your device. If you lose the device, you will lose the access to your accounts (all of them).
This was actually my surprise with the Google 2FA back in the day. Knowing it was Google, I automatically expected that the backup codes were somewhat synced to my Google account; hence when I downloaded and installed Google 2FA on my freshly factory restored mobile phone(without making a backup of the keys), well, let's just say I didn't have a pleasant experience trying to contact all the service representatives from 5+ accounts I had with 2FA activated..