I think you are a little confused with Bitwarden Authenticator. In the free version, you can actually supply your Bitwarden account (i.e. your password database) with 2FA via email or another Authenticator app. I do this, for example, with Microsoft Authenticator.

What you have to pay is to use Bitwarden itself as your TOTP generator, so you can also fill in 2FA from Bitwarden after autofilling your credentials (set it to be just a simple ‘paste’). This of course makes your pwd database a single point of failure because it also contains your 2FA tokens, but then you can also counter something by putting at least an external 2FA in your database and using long passwords generated with a little addition Segment (before or after the password Bitwarden generates for you) does not store in the database.

Anyway, I find it very easy to now enable two-factor authentication (2FA) wherever I can, even on websites which wouldn’t be a disaster if they got into my account.

If you don’t want to pay for it, you can also host Vaultwarden yourself, as mentioned above, or your own “Bitwarden server” on which you can use all functions for free.