I'm not sure what you mean by that.
2FA can't be linked back to your phone or anything if that is your concern. If you use it to login to your account often, then maybe the platform can.
If you want an alrernative, check platforms that support passkeys as a 2FA, it has the same privacy though.
Also, you don't specify which platform you're using. As @I_Am_Jacks_____@lemmings.world mentioned, Aegis for Android is great. For iOS, the pull request to include ente Auth in the Privacy Guides recommendation page for authenticator apps is already merged. It will reflect on the live website once the Privacy Guides team decides to release v3.17 of the site.
I can understand people never read boring privacy policy. But you should at least read the discussion you mention!? It is not recommended and the discussion about replacing Ravio, as it was good 2fa app for iOS, but was sold. Android has Aegis and a lot of other good apps, this one does not stand a chance. Anyeay, they use google analytics to collect data, which is really not needed for 2fa app at all and privacy wise any trackers are not good.
As long as you aren't backing up your tokens to the cloud they're all going to be functionally equivalent in terms of your data privacy outside of intentionally malicious apps. I mean that in the sense that no authenticator app should be sending your tokens anywhere on the internet. Use common sense when it comes to installing Google or Microsoft's authenticator apps.
If you like syncing across devices, Bitwarden (paid) or Proton Pass. If you already use Bitwarden and don't like the idea of having passwords and 2FA in the same place, just make a second Bitwarden account.