Lots of comments here suggesting forwarding emails from the external account to Gmail as a workaround for this.

This isn't a complete workaround though. In particular, the option to delete the email from your server after retrieving it will not be replaceable. At least cPanel doesn't seem to offer the option to delete automatically after forwarding, and you could argue it shouldn't - with a push, you never know if the other end actually got it, unlike with a pull.