Postmaster can definitely do this provided you use a mailing service, and not using the built-in communication module. You could however write an email service for Postmaster that reused your mailing list vs. deferring to MailChimp or something.
I advocate using MailChimp or CampaignMonitor over another services because sending email is otherwise unreliable. You will find that people simple don't receive some messages, even though the server sent them. PHP doesn't do email very well by default, so it's best to use one of these services. And to use these services with small amounts of email won't cost any money anyway.
Postmaster can handle all your template and allow you to use channel entries to create new campaigns. You can use whatever fieldtypes you desired, so the data entry portion is really great since clients are already use to it. The real challenge in my mind will be how to reuse the native mailing list module. Other users have requested support for it, but to honest, I can't even figure out how it works (and it's not very good anyway). If you decide to go with an email service, Postmaster includes all the tags needed for signup.
While Postmaster may seem like overkill, the template parsing isn't an easy task. All the options and configuration are really what make Postmaster as good as it is. For instance, you could setup Postmaster to send "draft" campaigns when an "open" (default) status, but only send to the real subscriber list with a "Publish" status. The possibilities are endless.