Earlier quoted context omitted.
But if a site is self signed the browser could show something like "This site is not authenticated, don't enter any personal info/passwords"
Right. It is questionable why self-signed certificates trigger scary warnings in browsers when totally unencrypted connections do not. I think the whole UI aspect of web transport security needs re-thinking.
The criteria is not self-signed, it's trusted/authenticated or not. Most self-signed certs are not trusted, and solving that solves the CA problem. But if a self-signed cert is trusted then browsers happily display the secure UI without any errors.