There is a simple solution to this. I'm a software engineer/designer by trade, so here's how I'd solve it without getting in the way of the elegance and simplicity of iOS.
Add one single feature available to any application:
"Share with other apps"
When you do that, a link to that data is made in your "Files" browser (which is available to apps that support it). Any app that is able to read that type of file, will then be able to see it when you access the "Files" section.
Problem solved. Steve, pay me.
Doesn't the iPad/iPhone already have this option? It's called "Open in..." and your apps can share these files with each other.