Currently I use Bitwarden on both my phone and my pc, but I'm looking into self hosting it with vaultwarden. This gives you access to premium features (such as TOTP support, for which I currently use Aegis Authenticator). It also gives you full control over your data.
If you start storing the password and 2FA at the same place, it defetes the purpose of the 2nd factor authentication. Better to keep them in two places.
Without the passwords the 2fa codes are nothing. Even if your password manager got hacked, most of your accounts would be safe as the 2fa is not there.
I was thinking about it, but unless you get your password manager compromised one-time passwords still offer additional protection (in case only the password is leaked in a security breach).
I will try self hosting regardless, but for now I might omit TOTP and stick with Aegis.
I do self host and it's worth it. I just have a Raspberry Pi in the corner next to the router running all the time and I've had no issues for about a month of doing it now.