It's actually much worse than you expect. Most email clients don't show the email address of the sender; they show the display name. If you get an email from "The President <[email protected]>", most users will see that the email comes "The President." And [email protected] could get a valid signature for her domain, because she obviously controls it, and people would see that the email from "The President" really did come from "The President."

There are many problems with email. The ability for [email protected] to claim to be [email protected] is not at the top of the list.