Thanks for the info Andrew. Cool story idea!
It is possible to embed configurable web apps in a story map, so you could use the Filter app or Web AppBuilder and embed it in your story to let your audience perform the filters you described. You'd do this just like you've already embedded Map Series apps within another Map Series app (using the process outlined here: Embedding a Story Map within a Story Map | ArcGIS Blog).
However, I always caution story map authors to make sure they really want to do this kind of thing. The best Story Maps are simple for the audience to use, and embedding GIS-style apps in a story can sometimes introduce complexity that negatively affects your audience's experience. If you are publishing this for a professional or tech-savvy audience then it is often ok to embed GIS apps, but if your audience is the "general public" or people in a non-technical field you may want to reconsider this approach.
The Map Journal app has a specific feature called main stage actions that helps you lead your audience through a story by changing the appearance of the map (toggling layers with filters applied, etc.) without expecting them to figure out how to use a query widget or other GIS tools. Rather than having a series of tabs with all the different years you could tell a story about why this issue is important and then include a section for roofs, HVAC, etc. and have main stage action links in your story narrative for each of the year ranges. When a reader clicks on a link the map would update to show those schools.
Then you could create some additional layers showing some of the filters you'd expect people to be interested in (like the one you described in your previous post). You could add one or more sections exploring these scenarios and give the audience your perspective on them since you are a subject matter expert in the data.
If you still wanted to give people the chance to explore the data you could include a final section in the Map Journal that has an embedded Web AppBuilder app with the query widget you described. You could explain how to use the app in the side panel narrative. This way you've already walked someone through the data and several examples and then you can set them loose to do some exploration on their own if they are interested.
Here's a blog post about main stage actions if you want to pursue that option: Using Story Map Journal Main Stage Actions | ArcGIS Blog
Hope this is helpful!
Owen
Owen Evans
Lead Product Engineer | StoryMaps