If you put cpanel on a server - not saying you should - but don't you get a complete mail setup with dovecot, postfix, spamassasin and the whole shebang? And then set it to relay smtp through mailgun.
Isn't that what all those website and email providers do? Or are they each running custom configs?
Typically they only offer IMAP and POP, even a web interface is rare. It's normally very basic email, not "groupware" like most business mean with calendars and stuff. So you have to bring your own IMAP client.