I'm going to disagree with your premise that it makes sense to keep some passwords out of a (secure and non-proprietary) password manager.
While it might seem like some passwords might make sense outside of it I think those might be limited to two types:
A full-disk encryption key on a device which contains your password manager itself. This is mostly because you'll need that password to get into the password manager itself. For generating this password use something secure and random like a 7-9 word diceware password.
The password to the password manager itself. Only because it is irrelevant to do so, not for any security purpose.
I'll also respond to your proposals:
Primary e-mail: If you're password manager is compromised you are most likely already screwed on the email login. Assuming a route of attack that let someone into your password database (keylogger, screen capture, whatever) you already have a significant vulnerability that almost definitely includes access to your email password. Why not just include your email password(s) for security purposes? On the note of two-factor auth problems it might just make sense to use another diceware password for your email so you have it memorized too. The cost benefit favors keeping your password in a secure database strongly.
Bank Accounts - Any bank that isn't stuck with its head in the sand should LOVE that you are using a password manager. Financial data should always be protected by randomized keys. I'm not sure what terms violations you would run if you did this. Especially if these are your personal accounts we're talking about. If they are other accounts with financial information that isn't yours I'd say you have an ethical (if not legal) obligation to protect the data as well as you can. This means using random generated passwords from a password database.
Access codes to physical hardware. While I understand the idea behind this, if you actually have to deal with physical hardware a lot and want secure codes you also want a secure method of storing the codes and making them strong. If you resort to using bad codes or the same mildly good code everywhere you'll end up doing more harm than good. If you need to, get a netbook or secure phone and put the password database on there.
You ask: "Is there a different trade-off than risk aggregation vs. convenience?"
I would encourage you to think of password managers less as convenience and more as tools to maintain good security. Breached password managers would indeed be a problem, but the problems they prevent are quite substantial and the likelihood of their failure with proper setup are minimal.