You and I are educated enough about web security that we know not to type one site's password into another site.

Many people aren't. Phishing is a real problem.

When "legitimate" sites start doing slimy, insecure things like asking for third-party passwords, three things happen. One, those "legitimate" sites have the power to do things that most users don't really want them to, like spam their entire contact list as them. Two, it becomes harder for unsophisticated users to distinguish legitimate sites from phishing sites. Three, it means that if a criminal breaks into a "legitimate" site, there's more valuable information there for him to steal.