Should is only your opinion here. There may be privacy concerns - for example, after I log out I may not want other people to know that I was visiting the site. Especially when the email address is used as login. And really important on banking sites, etc. Also, as Morris said, every modern browser remembers input form data, so no need to replicate that with a cookie. I am the one who doesn't like seeing my username filled in automatically after I log out of the site.
And that is a good thing and let it stay that way - I can control this behaviour in my browser settings and I don't want web sites to impose on me filling in my username automatically.
If all browsers did that automatically there'd be no need for any login forms to include a "remember me" checkbox.
There is a difference between a browser remembering my credentials and "remember me" function. When I rely on my browser then I get logged out each time I end my session and when I come back I need to log in again through a login form so each time I come back I get to choose whether I want to log in (in a single click since my browser fills in my credentials) or browse the site as a guest. When I check "remember me" then each time I come back I am already logged in, no need to go through the login form again.