You must send through SMTP servers when you send as example.com. However, this functionality is not available for your account. Please contact your domain administrator.
I used the Chat help option from Gmail and was helped by Dimitar who suggested the following fix:
Dimitar B: Gmail
Dimitar B: Advanced settings
Dimitar B: Under “Allow per-user outbound gateways” – Disable the checkbox and re-enable it again
Dimitar B: This should activate this setting although the downside of this is that it could take up to 24 hours to propagate in full