Sorry for being quiet. We've made progress, can reproduce the issue, and are working on trying some solutions.
Big shout out to Michael (Misha) who was two steps ahead of us and did incredible finding nuggets of information scattered on the internet.
Seems like a number of applications, virus programs, system apps, Steam (yes, fucking steam), changes a field in the system registry for every USB HID Device's "restricted" boolean from "false" to "true". Why? No idea. Uninstalling the application does not revert the registry change, which validates what some of you experienced. Steam can install advanced drivers to help support controllers like XBOX-360, and this is what changes the field for that specific app.
We are trying to create a version of the app that detects this, and updates the field. But this is not so simple. UWP apps don't have permission, and we have to do a run-around. But this seems possible, and the path of least resistance. We have some other ideas, but this seems like the best approach.
Un. Fucking. Believable.