unfortunately this is an iOS restriction. iOS doesn't allow apps to be able to access files from other apps, so a file picker is kinda useless right now because you would only be able to get files from the app.
We implemented a workaround for this. You need to open the file you want to upload using an external app (like Safari), and then use the "Open in..." button to open it with Moodle Mobile. Once you do it, the app should upload it to your private files.
I just checked in my iPad with iOS 9 and I can't see the "Open in" button in the Gallery, but I do in Safari. In the Gallery I only get the "Share" button, and that doesn't work with the app (see this issue). Can you please tell me if you're able to share files from the gallery to Moodle Mobile?