OTPs don't protect against phishing. You still type the TOTP in a browser window that sends it off to the attacker. Phishing SDKs automatically handle proxying the password over and then proxying the TOTP over.
On the topic of phishing and OTPs, storing the OTP in your password manager could actually help with phishing (opposed to storing it in an authenticator), because it will only autofill on the correct domain. This can be the difference between compromising a password or the whole account.