Hi Sterp,
I also checked sending mails. The fact is, that newsletters/mails of BwPostman will reach the receiver, if mails of Joomla! reach the receiver. If Joomla! has a problem with sending, BwPostman also as a problem with sending.
The process of sending mails is not exactly the same as of Joomla!. Joomla! also checks the connection to the mail server and gives a response, if configuration doesn't meet the provider needs. But the rest is the same.
Once again: If Joomla is able to send, BwPostman is able to send. Your screenshot shows, that BwPostman has sent the newsletter. If the newsletter does not reach the receiver, perhaps there is a typo in the mail address or the mail server of the receiver blocks the mails. I can't see another possibility.
Have You tried to subscribe to the mailing list in the front end by module or component? What is the result in this case? Do You get the confirmation mail?
Best
Romana