Thanks for the question. You are correct. Once an automator application is saved, it becomes a standalone application (no longer an automator “document”, so that it can run without the need for Automator).
The only way to edit your application is to open Automator first, then choose File > Open, and search your Mac for the application (do not search for automator documents, as it is now a standalone application. I personally keep all of my automator applications in a specific folder, for easy access later).