For me it depends how “hardened” the account needs to be.
For key accounts I keep the 2fa seperated (where possible with a fido token) For not so relevant accounts I add it to bitwarden. Because I probably wouldnt activate 2fa for these accounts if I’d have to pull out the phone each time
Edit: you also can add the 2fa token to a separate vault
The app is intentional, with browsers they can’t control which extensions you run, and therefore can’t force their ads on you. With the app they can control the environment and you are legally not allowed to modify their app because trademark…