In Pages, Numbers, and Keynote 3.3 for iOS, Apple has done away with its custom file manager and instead uses the a version of the iOS 11 Files App to allow the user to choose and save files within the applications,
The Files-like document picker User Interface on Pages and Keynote 3.3 for iOS 11:
Is this a publicly available view controller that developers can implement in 3rd-Party Applications?
Apple mentions in a user support document that these apps "work together" to present this view. Link: https://support.apple.com/en-us/HT208077
I believe it is UIDocumentBrowserViewController. It's essentially the successor to
As far as "how did they make it the root view controller of the app? Isn't [it] usually presented from another View Controller?"
Check this page: Adding a Document Browser to Your App.
Always assign the document browser as your app's root view controller. Don't place the document browser in a navigation controller, tab bar, or split view, and don't present the document browser modally.
There are a lot of useful resources on that page, but a lot seems to hinge on your app being "Document Based" (About Document-Based Applications in iOS).