1password and Lastpass both manage MFA codes and work cross-device. If your phone is your only way to get into MFA protected accounts, you're doing it wrong. What happens if you can't repair your phone?
I don't understand your point. This is exactly what multi-factor authentication protects against if you don't store your MFA codes in your password manager.
standard TOTP MFA (which is what most password managers would offer in terms of MFA) uses a shared secret, which you would just dump from the same database you get the dumped passwords from.
unless you use asymmetric crypto e.g. in webauthn this doesn't benefit you at all.
If your concern is phishing, storing the code on the device (especially on a modern phone) is really moot since all forms of one-time codes are vulnerable — you should be working on how to switch to FIDO2/WebAuthn.
If your concern is a temporary exploit of the user's browser, using an out-of-process password manager is likely to prevent exfiltration of the entire password list but in practice this is already probably a disaster scenario unless you're using sites which require a strong FIDO2 challenge for sensitive operations since the attacker already has your cookies for everything you use regularly.