Edit: If you already have two phones due to your work life separation, then 2FA isn't really causing you to get a second phone is it?
https://www.twilio.com/legal/privacy/authy
The only privacy friendly method is no 2FA, just long secure passwords. Which is why 2FA is pushed so hard ...
This is absolutely ridiculous and so clearly false I can't help but wonder if it's intentional misinformation.
There are numerous open source TOTP authenticators for both the desktop and mobile that require absolutely no data to be stored in the cloud or shared with third parties.
I myself use KeepassXC and Keepass2Android.
If I was really paranoid I'd keep the TOTP database on a separate device but, frankly, I don't anticipate being the target of a motivated attacker so that's more than I feel is necessary given the threat models I'm concerned about, those being untargeted hacks (service breaches, driveby attacks, etc) and social engineering.
If you don't want to use Authy then use something else that doesn't backup the 2FA codes for you. But don't say 2FA is inherently a privacy concern. It isn't.