But you're still "managing umpteen million passwords", hence the name password manager.
That makes no sense. You use a password manager so you don't need to manage the passwords. If you don't like the concept of having passwords at all then use a stateless, hash-based password generator instead.
How does one integrate that with Facebook or Google or Nameasite?
A browser extension could be made pretty easily. If you have a terminal open, do `echo mymasterpassword facebook.com | shasum -a 256 | pbcopy`.