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You wouldn't need to store the codes, but you would need to store the key that makes the codes.

If it's secured by TOTP and a unique and secure password, it's not the weakest link.

This reduces the attack scope from two devices to one. If your computer or web browser is compromised then both your TOTP secrets and password secrets are in one basket. Storing TOTP on a separate device can make it significantly harder to compromise your accounts.

He might be using two different password manager accounts, one for passwords and one TOTP? Although it doesn't help much if he logs in from the same machine anyway.

Partially, but it still offers protection against e.g. replay attacks and e-mail hacks.

I would say not when you can secure bitwarden with a hardware key 2FA.