I personally use Bitwarden for my 2FA needs. As others mentioned you can self host the server but personally I have no reason not to trust their SaaS solution, especially now that they offer EU hosted servers. If all you want is a basic authenticator app that does only one thing give FreeOTP a try, it’s made my RedHat. You can then sync the applications state.
That is true. But that’d be the case for any online/cloud 2FA service. So you could either have a local 2FA app just for Bitwarden or set up less secure but more convenient email 2FA.