Sadly no.

What is the simplest way to create an archive of previous emails sent through Shopify's email marketing system? I want to be able to let new subscribers read what I've written in the past. I've seen this done by Ryan Healy and other marketers, and wo...

I found a solution. When you set up your welcome sequence, simply add an internal email to yourself letting you know that you have a new subscriber. 

Agreed, please add notifications when someone subscribes. It's silly to add an app for this.