I was having the same problem and found the solution. The answer is the cPanel MX Entry setting.
By default it is set to Local Mail Exchanger in which case the web server tries to handle email that it believes are for it (i.e. email that originates from the same domain as the website).
To solve the problem, login to cPanel and change MX Entry to Remote Mail Exchanger in which case the web server will let the MX Records in your DNS to route the email.
Once you do this, there is no need for any Wordpress SMTP plugin at all.
