Yes, it is technically possible to bypass ArcGIS Online and reconfigure the storytelling templates to use an service from ArcGIS Server instead of building a web map. However, this requires the user to have JavaScript developer on hand to construct the map definition within the code of the app. Within every app, there is a function that loads the web map, if you switch this function to use a standard map through the JavaScript API and it matches the app required layers, it should just work.
Loading a web map through the API is basically the same as loading a standard map except the web map will bring across all the layer definitions, popups, etc. that were configured through ArcGIS Online.
We have decided to make all of our apps use web maps because they allow our users to configure various layer types without having to know JavaScript.