If you do care about privacy first things first you wouldn't be using apps provided by google itself. Although I think google authentication is okay, I guess you don;t still know what it can do in the background. You probably have some old mobile phone so I'll advise you to use that offline without connecting to the internet so you use that particular phone only for 2FA.

This eliminates the risks of somebody even compromising your phone and getting access to your 2FA codes and other stuff as most people like to keep their mails + 2FA on same phone and other information which can be a nightmare if somebody compromised your phone. We all know that mobile phones are somewhat first target for hackers as all of us have them and keep sensitive information in them, that's why I'll advise you to have separate mobile phone for 2FA only, if you don't have any old smartphone you can order some china phones for like 50bucks or even less they are able to run 2FA offline, all you need.