Yes a secondary/backup device gives a lot of peace of mind .Google Authenticator also makes it easy to duplicate all of your accounts at once to another device.
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And you don’t need to sync it to your Google account to do this. It just gives you a series of QR codes to scan with the new device, and then asks if you want to keep or delete them from the original device.
Oh yeah, forgot about that function, that's really neat as well for when you migrate to a new phone and have the old one still available. If I remember correctly, you can make a selection which ones you want to transfer through animated QRs.