Most sites that support authenticator apps support the TOTP standard that allows you to use any authenticator app. You don't have to install the app specific to the site, you can find a privacy-respecting one.
If I can find a privacy respecting one that's a good thing! I worked a job that tried to force us to use Microsoft authenticator but we pushed back after looking at the privacy policy and so instead we ended up with perfectly nice key fobs. It's hard to beat that for privacy.