I Don't recommend to use 2FA google authenticator as its's almost impossible to recover your account once you lost your codes. I've lost my device in which my codes were saved and i then i realized that i am unable to access my google account even my phone number was registered in gmail . I've lost all my data and trust me It sucks.
All of your logins protected by 2FA google authenticator have each a backup code that can be used in another device. And if you are looking for a more secure option, try Authy ; it's a nice alternative for google authenticator and can be secured by a single code backup which can be used to recover the app with all the accounts within if your device gets compromised or you accidentally erase the app.
About 2FA google authenticator, i didn't face problems using it but i remember hearing that it has been somehow compromised and some hackers succeeded to login users accounts using it. Not sure about this info so maybe someone could correct it .