If you don't keep your 2FA keys in your password manager, it offers much better protection than anything you can come up with in your head.

What happens when you need to change a password on one site? How do you know which password algorithm you're using on which site now?

Why am I asking you this? Because I was in the same situation and it became difficult to track. Now I keep all my passwords in a password manager, my 2FA codes in another app/service, and my recovery codes in the third place that's not connect with the first two in any way. Similarly, obviously, I don't have my 2FA service account + password in my password manager.