You don't go through the setup process on the sites again. The sites have no knowledge that you have 1 or 21 new totp apps set up. You just enter the saved seed keys into the app and it starts spitting out the same correct codes as the other apps you already had setup.

Gnome authenticator can export a json file containing the keys to all the sites you have in it. You can then take those (just manually read them in a text editor), and enter them into Google Authenticator on a phone, and now you have 2 working authenticator apps, both spitting out the same correct codes every 30 seconds.

Further, you take that same json and paste it into a note in a keepass record, or save the individual seed keys in individual site entries just like the passwords, and copy that keepass db file all over the place including cloud drives, and including places you can access without the totp.

Now you can reproduce a working authenticator from scratch on any device at any time no matter where you are and no matter what happens to your phone or laptop. Buy a brand new phone or laptop, have a way to get a copy of your keepass db without needing the totp app, and in a couple minutes you have a working totp app again.

You never really have to even use the single-use emergency bypass codes. Keeping copies of the initial setup seeds is really no different from keeping copies of the emergency codes, but the setup seeds reproduce a fully working app not just a one-time access to a site.

And even if some app doesn't provide an export like gnome authenticator, you can also just record the key the first time it is generated instead of just scanning the qr code. Once you've saved it, you can use it as many times as you want.