IMO it's good practice to not one's vital 2FA codes held hostage on a service where one's accounts or IP address can be flagged for spam or abusive behavior by automated systems. (This also applies to Google Authenticator, for what it's worth!) Especially in a world where customer service teams are being trimmed wherever possible!
I use 1Password - its UI leaves some things to be desired, and it's not cheap, but it has zero incentive to cancel the account of any paying customer!
I switched out of Google Authenticator when they updated the app and all my 2FA just went away.
They did fix it with another update, but that was a seriously un-fun few days. Luckily I was just logged in to my AWS account so I could disable the 2FA.
I use 1Password - its UI leaves some things to be desired, and it's not cheap, but it has zero incentive to cancel the account of any paying customer!