Improve FilePicker for better iOS support

[ewpatton]

Describe the desired feature

As discussed in #3212, the iOS APIs to expose files to the shared Files app are designed in a completely different way than the Android equivalent APIs. Unfortunately, we designed the FilePicker fairly tightly coupled to the Android design so certain behaviors of the FilePicker on iOS make no sense and cannot be implemented in the existing design. Rework the FilePicker design to be more broadly compatible with both platforms.