Here’s what I don’t get. How come all of the people who corresponded with her never noticed that her address didn’t end in “gov”?
Every person was different, and her aides were apparently go-betweens a lot of the time.
Depends upon your email client if it even shows the actual recipient address. Also, you can easily set up a 'vanity' domain that merely seems to be a private address, when in fact mail for the domain is handled by the appropriate party. That's what the MX (mail exchange) records are for in DNS, to let email systems know how to route mail. There should also be appropriate SPF DNS records to make everything work correctly.
All of that said, the vast majority of folks don't know anything about how any of this stuff works. Someone should have raised red flags about this while she was SoS. The fact that there were plenty of the media who were likely aware of what was going on, yet never raised the issue is an indictment of their willingness to cover for her.