1. 2FA.
2. Still require a (non-trivial) username to be manually entered with each login. Users would be educated to not store their usernames with their master password. (In essence usernames _are_ another password, if you keep yours private.)
3. A spin on the 2FA idea would be to have a multi-sig solution, where you have two security apps that need to sign to gain access, each stored separate from one another. For example, one could be stored in a secure cloud location and another on your desktop, or one on your phone and the other on a laptop.
I think that what you have in mind would make the whole password thing seamless and 'painless'.
Personally I use Keepass to keep all my strong passwords and this is linked to Chrome through the extension so it automatically logs into sites. Sites offering 2FA needs the additional code from either the google/auth apps on my phone. Keepass runs from a USB drive and the 2FA's on my phone so stored in 2 different locations.
This comes close to what you are explaining but it might be way too complex for the average Joe so your solution and or idea would be great as it would make the whole password process painless.
Thanks; how do you like the Keepass, any complaints with it? My only objection to solutions like it is that I'm a cheapskate and Keepass typically means carrying a USB drive. Hardware generally costs something, whereas software is often free (if you aren't picky).