I’m not sure what your problem is. I assume you are NOT using the free hosting service on byethost as it has limitations on email.
Although difficult to find, I did locate the email setup instructions for premium host user on byethost. It is consistent with what you have. However, the same documentation only shows PHP versions up to 5.6x. You are definitely on 7.2x. So it is possible that the knowledge base doc is out of date and there is another email setup option that you can use; possibly using TLS.
I suggest you talk with your web host support team. They might be able to look at their logs and see why your connection is timing out.
Other than that, I’m at a loss as to why you are having an issue.