This spells the end of privacy on the Internet. Your phone 100% identifies you. Plus, if you lose it, you may lose your account as well. I just got an email from Twitter support, saying that I could “restore” my lost access by creating new account. I thought this email was some kind of a joke, but no.
That is why I use a TOTP app that lets me export an encrypted copy of the TOTP secrets. I also have a printed-out copy of the secrets stored in a secure location, which I update monthly for new accounts I add in. Prior to that when I was just using Google Authenticator I would record the TOTP secret prior to importing it into the app.
What TOTP app do you use, and what format do you print the secrets in?
I currently use "OTP Auth" for iOS, which supports backup and encrypted cloud sync, but I'm not sure if that supports something like "export to a text file".
It is open source, maintained, easy to use, can do backups and re-present the QR code so you can easily scan it with another device.