My advice is to either spend time learning the Drupal / PHPList newsletter systems so you can maybe write your own step by step basic instructions for sendnig a newsletter, or make one yourself.
Unfortunately making one yourself, especially for clients, means it will take time to iron out bugs, and it will probably not have the same functionality as those already made and tested.
Almost everything I have done with websites involves seeing what already made stuff is out there, deciding it is too bulky or not user friendly, start making my own (for my specific needs), and finding that to get it up to the standard I want will take a lot more time and effort than I have, so I go back and learn the already made stuff.
You could make a very simple one for clients using a html form, TinyMCE (use Google) for the writing of the newsletter, and then I really recommend using a PHP mailer library like
SwiftMailer to send it (it just makes it easier in controlling your mail)*.
*You will need to know a bit of OOP (Object Orientated Programming), but you should learn it anyway, so don't let that scare you. Documentation is good and lots of people use it, so there is lots of support everywhere including here at Bytes.