So what happens if you lost the device and you were prohibited the registration on a new device? This means you can't log in even you have the login credentials. So what is the difference between Authy and Google Authentication? If you keep off prohibition then it would be a quite risky move, someone would guess your password or would be even hacked.
You can register 2 devices of yours. For example 1 smartphone and 1 notebook (as a back up). You will have all your 2FA codes in both devices, and you will disallow that someone add another device. For example, if a hacker steals your password, he won't be able to access any of your 2FA codes.
This is the difference between GA and Authy: You have the security that a hacker cannot add another device even if he steals your password, and you are still able to access your accounts if you lost one device.
If you lose your phone, just buy a new one and add that new phone from your allowed device.
Much better than GA.
The best solution is to maintain a notebook to write you all the accounts credentials including Google Authentication codes and keep it always under you. So even lost the device still you will have to backup all your account credentials.
Storing your passwords and private keys of 2FA codes in a file on a notebook is very dangerous (and those private keys are mostly very big and impossible to write down in a paper). This file contains sensitive information that a hacker can access without major difficulties and steal money that may exist on exchanges or other services where the user keeps cryptocurrencies.
Authy is a software created by cybersecurity professionals that know what they are doing. What you are suggesting is the same as storing bitcoin private keys in file in your notebook... This may work for a while, but you might (and probably will) get hacked one day.
Unless you are a security specialist, I would advice everyone to work with the appropriate software.