@Dougster I think that all system emails are sent by the function sendMailPhpMailer in file admin/lib.php around line 427
If you are sending through an smtp server then you can enable debug, which might show what is not working
add these lines
$mail->SMTPDebug = 3;
$mail->Debugoutput = 'html';
immediately after this line which is line 454
$mail = new PHPlistMailer('systemmessage', $destinationemail, false);
Then generate a system email, such as by adding a new subscriber using a subscribe page, while signed-in as an admin. There should then be some output showing the exchange with the smtp server.