Personally, I don't, I explicitly and intentionally keep my Email, Password Manager and 2FA providers all distinct and separate. (This was very useful for extracting the TOTP secret out of the Symantec VIP 2FA system: https://github.com/dlenski/python-vipaccess)

At work, we do for non-person accounts. It's non-ideal, but I don't want to manage two separate systems for storing that data and trying to keep them in sync. The password vault system itself has a hard requirement for 2FA to download the vault.

Convenience and security are constantly at war with each other.

I've decided that, for me, the risk of someone obtaining access to my password manager is very low. I've been using password managers since ~2000, randomly generated passwords; I've never had an account compromised.

There's enough other sensitive information in my password manager that if someone did obtain access to it, I'd be fucked anyway, to the point that someone getting my TOTP secrets would be minor.